YES

Problem:
 active(f(f(a()))) -> mark(f(g(f(a()))))
 active(g(X)) -> g(active(X))
 g(mark(X)) -> mark(g(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:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                [1 0 1]     [0]
   [mark](x0) = [0 0 1]x0 + [0]
                [0 0 0]     [1],
   
             [1 1 0]  
   [f](x0) = [0 0 1]x0
             [0 1 0]  ,
   
             [1 0 1]     [0]
   [g](x0) = [0 0 1]x0 + [0]
             [0 0 0]     [1],
   
                    
   [proper](x0) = x0
                    ,
   
         [0]
   [a] = [0]
         [1],
   
               [1 0 0]  
   [top](x0) = [0 0 0]x0
               [0 0 0]  ,
   
                
   [ok](x0) = x0
                ,
   
                    
   [active](x0) = x0
                    
  orientation:
                       [1]    [0]                     
   active(f(f(a()))) = [0] >= [0] = mark(f(g(f(a()))))
                       [1]    [1]                     
   
                  [1 0 1]    [0]    [1 0 1]    [0]               
   active(g(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = g(active(X))
                  [0 0 0]    [1]    [0 0 0]    [1]               
   
                [1 0 1]    [1]    [1 0 1]    [1]             
   g(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(g(X))
                [0 0 0]    [1]    [0 0 0]    [1]             
   
                  [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 1]    [0]    [1 0 1]    [0]               
   proper(g(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = g(proper(X))
                  [0 0 0]    [1]    [0 0 0]    [1]               
   
              [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 1]    [0]    [1 0 1]    [0]           
   g(ok(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = ok(g(X))
              [0 0 0]    [1]    [0 0 0]    [1]           
   
                  [1 0 1]     [1 0 0]                  
   top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X))
                  [0 0 0]     [0 0 0]                  
   
                [1 0 0]     [1 0 0]                  
   top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X))
                [0 0 0]     [0 0 0]                  
  problem:
   active(g(X)) -> g(active(X))
   g(mark(X)) -> mark(g(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=1
   
   interpretation:
    [mark](x0) = 2x0 + 1,
    
    [f](x0) = 2x0 + 1,
    
    [g](x0) = x0,
    
    [proper](x0) = 2x0 + 1,
    
    [a] = 2,
    
    [top](x0) = x0 + 6,
    
    [ok](x0) = 2x0 + 1,
    
    [active](x0) = 2x0
   orientation:
    active(g(X)) = 2X >= 2X = g(active(X))
    
    g(mark(X)) = 2X + 1 >= 2X + 1 = mark(g(X))
    
    proper(f(X)) = 4X + 3 >= 4X + 3 = f(proper(X))
    
    proper(a()) = 5 >= 5 = ok(a())
    
    proper(g(X)) = 2X + 1 >= 2X + 1 = g(proper(X))
    
    f(ok(X)) = 4X + 3 >= 4X + 3 = ok(f(X))
    
    g(ok(X)) = 2X + 1 >= 2X + 1 = ok(g(X))
    
    top(mark(X)) = 2X + 7 >= 2X + 7 = top(proper(X))
    
    top(ok(X)) = 2X + 7 >= 2X + 6 = top(active(X))
   problem:
    active(g(X)) -> g(active(X))
    g(mark(X)) -> mark(g(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))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [mark](x0) = 2x0 + 3,
     
     [f](x0) = x0,
     
     [g](x0) = 6x0,
     
     [proper](x0) = 2x0,
     
     [a] = 0,
     
     [top](x0) = x0 + 4,
     
     [ok](x0) = 2x0,
     
     [active](x0) = 5x0
    orientation:
     active(g(X)) = 30X >= 30X = g(active(X))
     
     g(mark(X)) = 12X + 18 >= 12X + 3 = mark(g(X))
     
     proper(f(X)) = 2X >= 2X = f(proper(X))
     
     proper(a()) = 0 >= 0 = ok(a())
     
     proper(g(X)) = 12X >= 12X = g(proper(X))
     
     f(ok(X)) = 2X >= 2X = ok(f(X))
     
     g(ok(X)) = 12X >= 12X = ok(g(X))
     
     top(mark(X)) = 2X + 7 >= 2X + 4 = top(proper(X))
    problem:
     active(g(X)) -> g(active(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))
    Matrix Interpretation Processor: dim=3
     
     interpretation:
                [1 1 0]     [0]
      [f](x0) = [0 0 0]x0 + [1]
                [1 0 0]     [0],
      
                     [0]
      [g](x0) = x0 + [0]
                     [1],
      
                     [1 0 0]     [1]
      [proper](x0) = [0 1 0]x0 + [0]
                     [0 1 1]     [0],
      
            [0]
      [a] = [1]
            [0],
      
                 [1 1 0]     [0]
      [ok](x0) = [0 0 0]x0 + [1]
                 [1 0 0]     [0],
      
                     [1 0 1]     [0]
      [active](x0) = [0 0 0]x0 + [1]
                     [0 0 1]     [0]
     orientation:
                     [1 0 1]    [1]    [1 0 1]    [0]               
      active(g(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = g(active(X))
                     [0 0 1]    [1]    [0 0 1]    [1]               
      
                     [1 1 0]    [1]    [1 1 0]    [1]               
      proper(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(proper(X))
                     [1 0 0]    [1]    [1 0 0]    [1]               
      
                    [1]    [1]          
      proper(a()) = [1] >= [1] = ok(a())
                    [1]    [0]          
      
                     [1 0 0]    [1]    [1 0 0]    [1]               
      proper(g(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = g(proper(X))
                     [0 1 1]    [1]    [0 1 1]    [1]               
      
                 [1 1 0]    [1]    [1 1 0]    [1]           
      f(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(f(X))
                 [1 1 0]    [0]    [1 1 0]    [0]           
      
                 [1 1 0]    [0]    [1 1 0]    [0]           
      g(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(g(X))
                 [1 0 0]    [1]    [1 0 0]    [0]           
     problem:
      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))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [f](x0) = x0,
       
       [g](x0) = 4x0 + 3,
       
       [proper](x0) = 5x0 + 4,
       
       [a] = 2,
       
       [ok](x0) = 4x0 + 6
      orientation:
       proper(f(X)) = 5X + 4 >= 5X + 4 = f(proper(X))
       
       proper(a()) = 14 >= 14 = ok(a())
       
       proper(g(X)) = 20X + 19 >= 20X + 19 = g(proper(X))
       
       f(ok(X)) = 4X + 6 >= 4X + 6 = ok(f(X))
       
       g(ok(X)) = 16X + 27 >= 16X + 18 = ok(g(X))
      problem:
       proper(f(X)) -> f(proper(X))
       proper(a()) -> ok(a())
       proper(g(X)) -> g(proper(X))
       f(ok(X)) -> ok(f(X))
      Matrix Interpretation Processor: dim=3
       
       interpretation:
                  [1 0 0]     [0]
        [f](x0) = [0 0 1]x0 + [0]
                  [0 1 0]     [1],
        
                  [1 0 0]     [0]
        [g](x0) = [0 0 1]x0 + [0]
                  [0 1 0]     [1],
        
                       [1 1 1]     [0]
        [proper](x0) = [0 0 1]x0 + [0]
                       [0 1 0]     [1],
        
              [0]
        [a] = [0]
              [1],
        
                   [1 0 0]     [0]
        [ok](x0) = [0 0 1]x0 + [0]
                   [0 1 0]     [1]
       orientation:
                       [1 1 1]    [1]    [1 1 1]    [0]               
        proper(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(proper(X))
                       [0 0 1]    [1]    [0 0 1]    [1]               
        
                      [1]    [0]          
        proper(a()) = [1] >= [1] = ok(a())
                      [1]    [1]          
        
                       [1 1 1]    [1]    [1 1 1]    [0]               
        proper(g(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = g(proper(X))
                       [0 0 1]    [1]    [0 0 1]    [1]               
        
                       [0]        [0]           
        f(ok(X)) = X + [1] >= X + [1] = ok(f(X))
                       [1]        [1]           
       problem:
        f(ok(X)) -> ok(f(X))
       Matrix Interpretation Processor: dim=3
        
        interpretation:
                   [1 1 0]  
         [f](x0) = [0 1 0]x0
                   [0 0 0]  ,
         
                    [1 0 0]     [0]
         [ok](x0) = [0 1 0]x0 + [1]
                    [0 0 0]     [0]
        orientation:
                    [1 1 0]    [1]    [1 1 0]    [0]           
         f(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(f(X))
                    [0 0 0]    [0]    [0 0 0]    [0]           
        problem:
         
        Qed