YES

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

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
               [1 0 0]     [1]
   [top](x0) = [0 0 0]x0 + [0]
               [0 0 0]     [0],
   
                
   [ok](x0) = x0
                ,
   
                    
   [proper](x0) = x0
                    ,
   
                  
   [mark](x0) = x0
                  ,
   
             [1 0 0]  
   [c](x0) = [0 0 0]x0
             [0 0 0]  ,
   
             [1 0 0]  
   [g](x0) = [0 0 1]x0
             [0 1 0]  ,
   
                    
   [active](x0) = x0
                    ,
   
             [1 1 0]  
   [f](x0) = [0 0 1]x0
             [0 1 0]  ,
   
         [0]
   [a] = [0]
         [1]
  orientation:
                       [1]    [0]                        
   active(f(f(a()))) = [0] >= [0] = mark(c(f(g(f(a())))))
                       [1]    [0]                        
   
                  [1 1 0]     [1 1 0]                
   active(f(X)) = [0 0 1]X >= [0 0 1]X = f(active(X))
                  [0 1 0]     [0 1 0]                
   
                  [1 0 0]     [1 0 0]                
   active(g(X)) = [0 0 1]X >= [0 0 1]X = g(active(X))
                  [0 1 0]     [0 1 0]                
   
                [1 1 0]     [1 1 0]              
   f(mark(X)) = [0 0 1]X >= [0 0 1]X = mark(f(X))
                [0 1 0]     [0 1 0]              
   
                [1 0 0]     [1 0 0]              
   g(mark(X)) = [0 0 1]X >= [0 0 1]X = mark(g(X))
                [0 1 0]     [0 1 0]              
   
                  [1 1 0]     [1 1 0]                
   proper(f(X)) = [0 0 1]X >= [0 0 1]X = f(proper(X))
                  [0 1 0]     [0 1 0]                
   
                 [0]    [0]          
   proper(a()) = [0] >= [0] = ok(a())
                 [1]    [1]          
   
                  [1 0 0]     [1 0 0]                
   proper(c(X)) = [0 0 0]X >= [0 0 0]X = c(proper(X))
                  [0 0 0]     [0 0 0]                
   
                  [1 0 0]     [1 0 0]                
   proper(g(X)) = [0 0 1]X >= [0 0 1]X = g(proper(X))
                  [0 1 0]     [0 1 0]                
   
              [1 1 0]     [1 1 0]            
   f(ok(X)) = [0 0 1]X >= [0 0 1]X = ok(f(X))
              [0 1 0]     [0 1 0]            
   
              [1 0 0]     [1 0 0]            
   c(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(c(X))
              [0 0 0]     [0 0 0]            
   
              [1 0 0]     [1 0 0]            
   g(ok(X)) = [0 0 1]X >= [0 0 1]X = ok(g(X))
              [0 1 0]     [0 1 0]            
   
                  [1 0 0]    [1]    [1 0 0]    [1]                 
   top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X))
                  [0 0 0]    [0]    [0 0 0]    [0]                 
   
                [1 0 0]    [1]    [1 0 0]    [1]                 
   top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X))
                [0 0 0]    [0]    [0 0 0]    [0]                 
  problem:
   active(f(X)) -> f(active(X))
   active(g(X)) -> g(active(X))
   f(mark(X)) -> mark(f(X))
   g(mark(X)) -> mark(g(X))
   proper(f(X)) -> f(proper(X))
   proper(a()) -> ok(a())
   proper(c(X)) -> c(proper(X))
   proper(g(X)) -> g(proper(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [top](x0) = 3x0 + 1,
    
    [ok](x0) = x0 + 4,
    
    [proper](x0) = 4x0 + 1,
    
    [mark](x0) = 4x0 + 1,
    
    [c](x0) = x0,
    
    [g](x0) = x0,
    
    [active](x0) = x0 + 1,
    
    [f](x0) = x0,
    
    [a] = 1
   orientation:
    active(f(X)) = X + 1 >= X + 1 = f(active(X))
    
    active(g(X)) = X + 1 >= X + 1 = g(active(X))
    
    f(mark(X)) = 4X + 1 >= 4X + 1 = mark(f(X))
    
    g(mark(X)) = 4X + 1 >= 4X + 1 = mark(g(X))
    
    proper(f(X)) = 4X + 1 >= 4X + 1 = f(proper(X))
    
    proper(a()) = 5 >= 5 = ok(a())
    
    proper(c(X)) = 4X + 1 >= 4X + 1 = c(proper(X))
    
    proper(g(X)) = 4X + 1 >= 4X + 1 = g(proper(X))
    
    f(ok(X)) = X + 4 >= X + 4 = ok(f(X))
    
    c(ok(X)) = X + 4 >= X + 4 = ok(c(X))
    
    g(ok(X)) = X + 4 >= X + 4 = ok(g(X))
    
    top(mark(X)) = 12X + 4 >= 12X + 4 = top(proper(X))
    
    top(ok(X)) = 3X + 13 >= 3X + 4 = top(active(X))
   problem:
    active(f(X)) -> f(active(X))
    active(g(X)) -> g(active(X))
    f(mark(X)) -> mark(f(X))
    g(mark(X)) -> mark(g(X))
    proper(f(X)) -> f(proper(X))
    proper(a()) -> ok(a())
    proper(c(X)) -> c(proper(X))
    proper(g(X)) -> g(proper(X))
    f(ok(X)) -> ok(f(X))
    c(ok(X)) -> ok(c(X))
    g(ok(X)) -> ok(g(X))
    top(mark(X)) -> top(proper(X))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [top](x0) = 4x0 + 1,
     
     [ok](x0) = x0,
     
     [proper](x0) = 2x0 + 1,
     
     [mark](x0) = 2x0 + 1,
     
     [c](x0) = 4x0 + 3,
     
     [g](x0) = 5x0 + 4,
     
     [active](x0) = x0,
     
     [f](x0) = 6x0 + 5,
     
     [a] = 6
    orientation:
     active(f(X)) = 6X + 5 >= 6X + 5 = f(active(X))
     
     active(g(X)) = 5X + 4 >= 5X + 4 = g(active(X))
     
     f(mark(X)) = 12X + 11 >= 12X + 11 = mark(f(X))
     
     g(mark(X)) = 10X + 9 >= 10X + 9 = mark(g(X))
     
     proper(f(X)) = 12X + 11 >= 12X + 11 = f(proper(X))
     
     proper(a()) = 13 >= 6 = ok(a())
     
     proper(c(X)) = 8X + 7 >= 8X + 7 = c(proper(X))
     
     proper(g(X)) = 10X + 9 >= 10X + 9 = g(proper(X))
     
     f(ok(X)) = 6X + 5 >= 6X + 5 = ok(f(X))
     
     c(ok(X)) = 4X + 3 >= 4X + 3 = ok(c(X))
     
     g(ok(X)) = 5X + 4 >= 5X + 4 = ok(g(X))
     
     top(mark(X)) = 8X + 5 >= 8X + 5 = top(proper(X))
    problem:
     active(f(X)) -> f(active(X))
     active(g(X)) -> g(active(X))
     f(mark(X)) -> mark(f(X))
     g(mark(X)) -> mark(g(X))
     proper(f(X)) -> f(proper(X))
     proper(c(X)) -> c(proper(X))
     proper(g(X)) -> g(proper(X))
     f(ok(X)) -> ok(f(X))
     c(ok(X)) -> ok(c(X))
     g(ok(X)) -> ok(g(X))
     top(mark(X)) -> top(proper(X))
    DP Processor:
     DPs:
      active#(f(X)) -> active#(X)
      active#(f(X)) -> f#(active(X))
      active#(g(X)) -> active#(X)
      active#(g(X)) -> g#(active(X))
      f#(mark(X)) -> f#(X)
      g#(mark(X)) -> g#(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))
      f#(ok(X)) -> f#(X)
      c#(ok(X)) -> c#(X)
      g#(ok(X)) -> g#(X)
      top#(mark(X)) -> proper#(X)
      top#(mark(X)) -> top#(proper(X))
     TRS:
      active(f(X)) -> f(active(X))
      active(g(X)) -> g(active(X))
      f(mark(X)) -> mark(f(X))
      g(mark(X)) -> mark(g(X))
      proper(f(X)) -> f(proper(X))
      proper(c(X)) -> c(proper(X))
      proper(g(X)) -> g(proper(X))
      f(ok(X)) -> ok(f(X))
      c(ok(X)) -> ok(c(X))
      g(ok(X)) -> ok(g(X))
      top(mark(X)) -> top(proper(X))
     TDG Processor:
      DPs:
       active#(f(X)) -> active#(X)
       active#(f(X)) -> f#(active(X))
       active#(g(X)) -> active#(X)
       active#(g(X)) -> g#(active(X))
       f#(mark(X)) -> f#(X)
       g#(mark(X)) -> g#(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))
       f#(ok(X)) -> f#(X)
       c#(ok(X)) -> c#(X)
       g#(ok(X)) -> g#(X)
       top#(mark(X)) -> proper#(X)
       top#(mark(X)) -> top#(proper(X))
      TRS:
       active(f(X)) -> f(active(X))
       active(g(X)) -> g(active(X))
       f(mark(X)) -> mark(f(X))
       g(mark(X)) -> mark(g(X))
       proper(f(X)) -> f(proper(X))
       proper(c(X)) -> c(proper(X))
       proper(g(X)) -> g(proper(X))
       f(ok(X)) -> ok(f(X))
       c(ok(X)) -> ok(c(X))
       g(ok(X)) -> ok(g(X))
       top(mark(X)) -> top(proper(X))
      graph:
       top#(mark(X)) -> top#(proper(X)) ->
       top#(mark(X)) -> top#(proper(X))
       top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
       top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
       top#(mark(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
       top#(mark(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X))
       top#(mark(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X)
       top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X))
       top#(mark(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X)
       c#(ok(X)) -> c#(X) -> c#(ok(X)) -> c#(X)
       proper#(c(X)) -> c#(proper(X)) -> c#(ok(X)) -> c#(X)
       proper#(c(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
       proper#(c(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
       proper#(c(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X))
       proper#(c(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X)
       proper#(c(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X))
       proper#(c(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X)
       proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
       proper#(g(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
       proper#(g(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X))
       proper#(g(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X)
       proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X))
       proper#(g(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X)
       proper#(g(X)) -> g#(proper(X)) -> g#(ok(X)) -> g#(X)
       proper#(g(X)) -> g#(proper(X)) -> g#(mark(X)) -> g#(X)
       proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> g#(proper(X))
       proper#(f(X)) -> proper#(X) -> proper#(g(X)) -> proper#(X)
       proper#(f(X)) -> proper#(X) -> proper#(c(X)) -> c#(proper(X))
       proper#(f(X)) -> proper#(X) -> proper#(c(X)) -> proper#(X)
       proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> f#(proper(X))
       proper#(f(X)) -> proper#(X) -> proper#(f(X)) -> proper#(X)
       proper#(f(X)) -> f#(proper(X)) -> f#(ok(X)) -> f#(X)
       proper#(f(X)) -> f#(proper(X)) -> f#(mark(X)) -> f#(X)
       g#(ok(X)) -> g#(X) -> g#(ok(X)) -> g#(X)
       g#(ok(X)) -> g#(X) -> g#(mark(X)) -> g#(X)
       g#(mark(X)) -> g#(X) -> g#(ok(X)) -> g#(X)
       g#(mark(X)) -> g#(X) -> g#(mark(X)) -> g#(X)
       f#(ok(X)) -> f#(X) -> f#(ok(X)) -> f#(X)
       f#(ok(X)) -> f#(X) -> f#(mark(X)) -> f#(X)
       f#(mark(X)) -> f#(X) -> f#(ok(X)) -> f#(X)
       f#(mark(X)) -> f#(X) -> f#(mark(X)) -> f#(X)
       active#(g(X)) -> g#(active(X)) -> g#(ok(X)) -> g#(X)
       active#(g(X)) -> g#(active(X)) -> g#(mark(X)) -> g#(X)
       active#(g(X)) -> active#(X) -> active#(g(X)) -> g#(active(X))
       active#(g(X)) -> active#(X) -> active#(g(X)) -> active#(X)
       active#(g(X)) -> active#(X) -> active#(f(X)) -> f#(active(X))
       active#(g(X)) -> active#(X) -> active#(f(X)) -> active#(X)
       active#(f(X)) -> f#(active(X)) -> f#(ok(X)) -> f#(X)
       active#(f(X)) -> f#(active(X)) -> f#(mark(X)) -> f#(X)
       active#(f(X)) -> active#(X) -> active#(g(X)) -> g#(active(X))
       active#(f(X)) -> active#(X) -> active#(g(X)) -> active#(X)
       active#(f(X)) -> active#(X) -> active#(f(X)) -> f#(active(X))
       active#(f(X)) -> active#(X) -> active#(f(X)) -> active#(X)
      SCC Processor:
       #sccs: 6
       #rules: 11
       #arcs: 52/289
       DPs:
        active#(g(X)) -> active#(X)
        active#(f(X)) -> active#(X)
       TRS:
        active(f(X)) -> f(active(X))
        active(g(X)) -> g(active(X))
        f(mark(X)) -> mark(f(X))
        g(mark(X)) -> mark(g(X))
        proper(f(X)) -> f(proper(X))
        proper(c(X)) -> c(proper(X))
        proper(g(X)) -> g(proper(X))
        f(ok(X)) -> ok(f(X))
        c(ok(X)) -> ok(c(X))
        g(ok(X)) -> ok(g(X))
        top(mark(X)) -> top(proper(X))
       Subterm Criterion Processor:
        simple projection:
         pi(active#) = 0
        problem:
         DPs:
          
         TRS:
          active(f(X)) -> f(active(X))
          active(g(X)) -> g(active(X))
          f(mark(X)) -> mark(f(X))
          g(mark(X)) -> mark(g(X))
          proper(f(X)) -> f(proper(X))
          proper(c(X)) -> c(proper(X))
          proper(g(X)) -> g(proper(X))
          f(ok(X)) -> ok(f(X))
          c(ok(X)) -> ok(c(X))
          g(ok(X)) -> ok(g(X))
          top(mark(X)) -> top(proper(X))
        Qed
       
       DPs:
        top#(mark(X)) -> top#(proper(X))
       TRS:
        active(f(X)) -> f(active(X))
        active(g(X)) -> g(active(X))
        f(mark(X)) -> mark(f(X))
        g(mark(X)) -> mark(g(X))
        proper(f(X)) -> f(proper(X))
        proper(c(X)) -> c(proper(X))
        proper(g(X)) -> g(proper(X))
        f(ok(X)) -> ok(f(X))
        c(ok(X)) -> ok(c(X))
        g(ok(X)) -> ok(g(X))
        top(mark(X)) -> top(proper(X))
       CDG Processor:
        DPs:
         top#(mark(X)) -> top#(proper(X))
        TRS:
         active(f(X)) -> f(active(X))
         active(g(X)) -> g(active(X))
         f(mark(X)) -> mark(f(X))
         g(mark(X)) -> mark(g(X))
         proper(f(X)) -> f(proper(X))
         proper(c(X)) -> c(proper(X))
         proper(g(X)) -> g(proper(X))
         f(ok(X)) -> ok(f(X))
         c(ok(X)) -> ok(c(X))
         g(ok(X)) -> ok(g(X))
         top(mark(X)) -> top(proper(X))
        graph:
         
        Qed
       
       DPs:
        proper#(f(X)) -> proper#(X)
        proper#(c(X)) -> proper#(X)
        proper#(g(X)) -> proper#(X)
       TRS:
        active(f(X)) -> f(active(X))
        active(g(X)) -> g(active(X))
        f(mark(X)) -> mark(f(X))
        g(mark(X)) -> mark(g(X))
        proper(f(X)) -> f(proper(X))
        proper(c(X)) -> c(proper(X))
        proper(g(X)) -> g(proper(X))
        f(ok(X)) -> ok(f(X))
        c(ok(X)) -> ok(c(X))
        g(ok(X)) -> ok(g(X))
        top(mark(X)) -> top(proper(X))
       Subterm Criterion Processor:
        simple projection:
         pi(proper#) = 0
        problem:
         DPs:
          
         TRS:
          active(f(X)) -> f(active(X))
          active(g(X)) -> g(active(X))
          f(mark(X)) -> mark(f(X))
          g(mark(X)) -> mark(g(X))
          proper(f(X)) -> f(proper(X))
          proper(c(X)) -> c(proper(X))
          proper(g(X)) -> g(proper(X))
          f(ok(X)) -> ok(f(X))
          c(ok(X)) -> ok(c(X))
          g(ok(X)) -> ok(g(X))
          top(mark(X)) -> top(proper(X))
        Qed
       
       DPs:
        g#(mark(X)) -> g#(X)
        g#(ok(X)) -> g#(X)
       TRS:
        active(f(X)) -> f(active(X))
        active(g(X)) -> g(active(X))
        f(mark(X)) -> mark(f(X))
        g(mark(X)) -> mark(g(X))
        proper(f(X)) -> f(proper(X))
        proper(c(X)) -> c(proper(X))
        proper(g(X)) -> g(proper(X))
        f(ok(X)) -> ok(f(X))
        c(ok(X)) -> ok(c(X))
        g(ok(X)) -> ok(g(X))
        top(mark(X)) -> top(proper(X))
       Subterm Criterion Processor:
        simple projection:
         pi(g#) = 0
        problem:
         DPs:
          
         TRS:
          active(f(X)) -> f(active(X))
          active(g(X)) -> g(active(X))
          f(mark(X)) -> mark(f(X))
          g(mark(X)) -> mark(g(X))
          proper(f(X)) -> f(proper(X))
          proper(c(X)) -> c(proper(X))
          proper(g(X)) -> g(proper(X))
          f(ok(X)) -> ok(f(X))
          c(ok(X)) -> ok(c(X))
          g(ok(X)) -> ok(g(X))
          top(mark(X)) -> top(proper(X))
        Qed
       
       DPs:
        c#(ok(X)) -> c#(X)
       TRS:
        active(f(X)) -> f(active(X))
        active(g(X)) -> g(active(X))
        f(mark(X)) -> mark(f(X))
        g(mark(X)) -> mark(g(X))
        proper(f(X)) -> f(proper(X))
        proper(c(X)) -> c(proper(X))
        proper(g(X)) -> g(proper(X))
        f(ok(X)) -> ok(f(X))
        c(ok(X)) -> ok(c(X))
        g(ok(X)) -> ok(g(X))
        top(mark(X)) -> top(proper(X))
       Subterm Criterion Processor:
        simple projection:
         pi(c#) = 0
        problem:
         DPs:
          
         TRS:
          active(f(X)) -> f(active(X))
          active(g(X)) -> g(active(X))
          f(mark(X)) -> mark(f(X))
          g(mark(X)) -> mark(g(X))
          proper(f(X)) -> f(proper(X))
          proper(c(X)) -> c(proper(X))
          proper(g(X)) -> g(proper(X))
          f(ok(X)) -> ok(f(X))
          c(ok(X)) -> ok(c(X))
          g(ok(X)) -> ok(g(X))
          top(mark(X)) -> top(proper(X))
        Qed
       
       DPs:
        f#(mark(X)) -> f#(X)
        f#(ok(X)) -> f#(X)
       TRS:
        active(f(X)) -> f(active(X))
        active(g(X)) -> g(active(X))
        f(mark(X)) -> mark(f(X))
        g(mark(X)) -> mark(g(X))
        proper(f(X)) -> f(proper(X))
        proper(c(X)) -> c(proper(X))
        proper(g(X)) -> g(proper(X))
        f(ok(X)) -> ok(f(X))
        c(ok(X)) -> ok(c(X))
        g(ok(X)) -> ok(g(X))
        top(mark(X)) -> top(proper(X))
       Subterm Criterion Processor:
        simple projection:
         pi(f#) = 0
        problem:
         DPs:
          
         TRS:
          active(f(X)) -> f(active(X))
          active(g(X)) -> g(active(X))
          f(mark(X)) -> mark(f(X))
          g(mark(X)) -> mark(g(X))
          proper(f(X)) -> f(proper(X))
          proper(c(X)) -> c(proper(X))
          proper(g(X)) -> g(proper(X))
          f(ok(X)) -> ok(f(X))
          c(ok(X)) -> ok(c(X))
          g(ok(X)) -> ok(g(X))
          top(mark(X)) -> top(proper(X))
        Qed