YES

Problem:
 active(c()) -> mark(f(g(c())))
 active(f(g(X))) -> mark(g(X))
 proper(c()) -> ok(c())
 proper(f(X)) -> f(proper(X))
 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]  
   [top](x0) = [0 0 0]x0
               [0 0 0]  ,
   
              [1 0 1]  
   [ok](x0) = [0 0 0]x0
              [0 1 0]  ,
   
                  [1 0 0]  
   [proper](x0) = [0 0 0]x0
                  [0 1 0]  ,
   
                [1 1 1]  
   [mark](x0) = [0 0 0]x0
                [0 0 0]  ,
   
             [1 1 1]  
   [f](x0) = [0 0 0]x0
             [0 0 0]  ,
   
             [1 0 0]  
   [g](x0) = [0 0 0]x0
             [0 0 0]  ,
   
                  [1 1 0]     [0]
   [active](x0) = [0 0 0]x0 + [1]
                  [0 0 0]     [0],
   
         [0]
   [c] = [1]
         [0]
  orientation:
                 [1]    [0]                  
   active(c()) = [1] >= [0] = mark(f(g(c())))
                 [0]    [0]                  
   
                     [1 0 0]    [0]    [1 0 0]              
   active(f(g(X))) = [0 0 0]X + [1] >= [0 0 0]X = mark(g(X))
                     [0 0 0]    [0]    [0 0 0]              
   
                 [0]    [0]          
   proper(c()) = [0] >= [0] = ok(c())
                 [1]    [1]          
   
                  [1 1 1]     [1 1 0]                
   proper(f(X)) = [0 0 0]X >= [0 0 0]X = f(proper(X))
                  [0 0 0]     [0 0 0]                
   
                  [1 0 0]     [1 0 0]                
   proper(g(X)) = [0 0 0]X >= [0 0 0]X = g(proper(X))
                  [0 0 0]     [0 0 0]                
   
              [1 1 1]     [1 1 1]            
   f(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(f(X))
              [0 0 0]     [0 0 0]            
   
              [1 0 1]     [1 0 0]            
   g(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(g(X))
              [0 0 0]     [0 0 0]            
   
                  [1 1 1]     [1 1 0]                  
   top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X))
                  [0 0 0]     [0 0 0]                  
   
                [1 1 1]     [1 1 0]                  
   top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X))
                [0 0 0]     [0 0 0]                  
  problem:
   active(f(g(X))) -> mark(g(X))
   proper(c()) -> ok(c())
   proper(f(X)) -> f(proper(X))
   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=3
   
   interpretation:
                [1 0 1]     [1]
    [top](x0) = [1 1 1]x0 + [0]
                [0 1 0]     [0],
    
                 
    [ok](x0) = x0
                 ,
    
                   [1 1 0]     [0]
    [proper](x0) = [0 0 0]x0 + [1]
                   [0 0 1]     [0],
    
                 [1 0 1]     [0]
    [mark](x0) = [0 0 0]x0 + [1]
                 [0 1 0]     [0],
    
              [1 1 0]     [0]
    [f](x0) = [0 0 0]x0 + [1]
              [0 0 0]     [1],
    
              [1 1 0]     [0]
    [g](x0) = [0 0 0]x0 + [1]
              [0 0 0]     [0],
    
                     
    [active](x0) = x0
                     ,
    
          [0]
    [c] = [0]
          [0]
   orientation:
                      [1 1 0]    [1]    [1 1 0]    [0]             
    active(f(g(X))) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(g(X))
                      [0 0 0]    [1]    [0 0 0]    [1]             
    
                  [0]    [0]          
    proper(c()) = [1] >= [0] = ok(c())
                  [0]    [0]          
    
                   [1 1 0]    [1]    [1 1 0]    [1]               
    proper(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(proper(X))
                   [0 0 0]    [1]    [0 0 0]    [1]               
    
                   [1 1 0]    [1]    [1 1 0]    [1]               
    proper(g(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = g(proper(X))
                   [0 0 0]    [0]    [0 0 0]    [0]               
    
               [1 1 0]    [0]    [1 1 0]    [0]           
    f(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(f(X))
               [0 0 0]    [1]    [0 0 0]    [1]           
    
               [1 1 0]    [0]    [1 1 0]    [0]           
    g(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(g(X))
               [0 0 0]    [0]    [0 0 0]    [0]           
    
                   [1 1 1]    [1]    [1 1 1]    [1]                 
    top(mark(X)) = [1 1 1]X + [1] >= [1 1 1]X + [1] = top(proper(X))
                   [0 0 0]    [1]    [0 0 0]    [1]                 
    
                 [1 0 1]    [1]    [1 0 1]    [1]                 
    top(ok(X)) = [1 1 1]X + [0] >= [1 1 1]X + [0] = top(active(X))
                 [0 1 0]    [0]    [0 1 0]    [0]                 
   problem:
    proper(c()) -> ok(c())
    proper(f(X)) -> f(proper(X))
    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=3
    
    interpretation:
                 [1 0 0]  
     [top](x0) = [0 0 1]x0
                 [0 1 1]  ,
     
                [1 0 0]  
     [ok](x0) = [0 0 0]x0
                [1 0 1]  ,
     
                    [1 0 1]  
     [proper](x0) = [0 1 0]x0
                    [0 0 1]  ,
     
                  [1 0 1]     [1]
     [mark](x0) = [0 1 0]x0 + [1]
                  [0 0 1]     [0],
     
               [1 0 0]     [0]
     [f](x0) = [0 0 0]x0 + [1]
               [0 0 1]     [1],
     
                    [0]
     [g](x0) = x0 + [1]
                    [1],
     
                    [1 0 0]  
     [active](x0) = [0 0 0]x0
                    [1 0 1]  ,
     
           [0]
     [c] = [0]
           [1]
    orientation:
                   [1]    [0]          
     proper(c()) = [0] >= [0] = ok(c())
                   [1]    [1]          
     
                    [1 0 1]    [1]    [1 0 1]    [0]               
     proper(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(proper(X))
                    [0 0 1]    [1]    [0 0 1]    [1]               
     
                    [1 0 1]    [1]    [1 0 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]               
     
                [1 0 0]    [0]    [1 0 0]    [0]           
     f(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(f(X))
                [1 0 1]    [1]    [1 0 1]    [1]           
     
                [1 0 0]    [0]    [1 0 0]    [0]           
     g(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [0] = ok(g(X))
                [1 0 1]    [1]    [1 0 1]    [1]           
     
                    [1 0 1]    [1]    [1 0 1]                  
     top(mark(X)) = [0 0 1]X + [0] >= [0 0 1]X = top(proper(X))
                    [0 1 1]    [1]    [0 1 1]                  
     
                  [1 0 0]     [1 0 0]                  
     top(ok(X)) = [1 0 1]X >= [1 0 1]X = top(active(X))
                  [1 0 1]     [1 0 1]                  
    problem:
     f(ok(X)) -> ok(f(X))
     g(ok(X)) -> ok(g(X))
     top(ok(X)) -> top(active(X))
    Matrix Interpretation Processor: dim=3
     
     interpretation:
                  [1 0 0]  
      [top](x0) = [0 1 0]x0
                  [0 0 0]  ,
      
                 [1 0 0]     [0]
      [ok](x0) = [0 0 1]x0 + [0]
                 [0 1 0]     [1],
      
                [1 1 1]     [0]
      [f](x0) = [0 0 1]x0 + [0]
                [0 1 0]     [1],
      
                [1 1 1]  
      [g](x0) = [0 1 1]x0
                [0 1 1]  ,
      
                     [1 0 0]  
      [active](x0) = [0 0 0]x0
                     [0 0 0]  
     orientation:
                 [1 1 1]    [1]    [1 1 1]    [0]           
      f(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(f(X))
                 [0 0 1]    [1]    [0 0 1]    [1]           
      
                 [1 1 1]    [1]    [1 1 1]    [0]           
      g(ok(X)) = [0 1 1]X + [1] >= [0 1 1]X + [0] = ok(g(X))
                 [0 1 1]    [1]    [0 1 1]    [1]           
      
                   [1 0 0]     [1 0 0]                  
      top(ok(X)) = [0 0 1]X >= [0 0 0]X = top(active(X))
                   [0 0 0]     [0 0 0]                  
     problem:
      top(ok(X)) -> top(active(X))
     Matrix Interpretation Processor: dim=3
      
      interpretation:
                   [1 0 0]  
       [top](x0) = [0 0 0]x0
                   [0 0 0]  ,
       
                  [1 0 0]     [1]
       [ok](x0) = [0 0 0]x0 + [0]
                  [0 0 0]     [0],
       
                      [1 0 0]  
       [active](x0) = [0 0 0]x0
                      [0 0 0]  
      orientation:
                    [1 0 0]    [1]    [1 0 0]                  
       top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X = top(active(X))
                    [0 0 0]    [0]    [0 0 0]                  
      problem:
       
      Qed