YES

Problem:
 active(f(x)) -> mark(f(f(x)))
 chk(no(f(x))) -> f(chk(mat(f(f(f(f(f(f(f(f(f(f(X())))))))))),x)))
 mat(f(x),f(y())) -> f(mat(x,y()))
 chk(no(c())) -> active(c())
 mat(f(x),c()) -> no(c())
 f(active(x)) -> active(f(x))
 f(no(x)) -> no(f(x))
 f(mark(x)) -> mark(f(x))
 tp(mark(x)) -> tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X())))))))))),x)))

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
               [1 0 1]     [0]
   [chk](x0) = [0 0 0]x0 + [0]
               [0 0 0]     [1],
   
         [0]
   [c] = [1]
         [0],
   
                  [1 0 0]     [1]
   [active](x0) = [0 0 0]x0 + [0]
                  [0 1 0]     [0],
   
                   [0]
   [no](x0) = x0 + [0]
                   [1],
   
         [0]
   [X] = [0]
         [0],
   
             [1 0 0]  
   [f](x0) = [0 0 0]x0
             [0 1 1]  ,
   
         [0]
   [y] = [0]
         [0],
   
              [1 0 0]  
   [tp](x0) = [0 0 0]x0
              [0 0 0]  ,
   
                   [1 0 0]     [1 0 0]  
   [mat](x0, x1) = [0 0 0]x0 + [0 1 0]x1
                   [0 0 0]     [0 1 0]  ,
   
                [1 1 0]     [1]
   [mark](x0) = [0 1 0]x0 + [0]
                [0 0 0]     [0]
  orientation:
                  [1 0 0]    [1]    [1 0 0]    [1]                
   active(f(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = mark(f(f(x)))
                  [0 0 0]    [0]    [0 0 0]    [0]                
   
                   [1 1 1]    [1]    [1 1 0]    [0]                                                   
   chk(no(f(x))) = [0 0 0]x + [0] >= [0 0 0]x + [0] = f(chk(mat(f(f(f(f(f(f(f(f(f(f(X())))))))))),x)))
                   [0 0 0]    [1]    [0 0 0]    [1]                                                   
   
                      [1 0 0]     [1 0 0]                 
   mat(f(x),f(y())) = [0 0 0]x >= [0 0 0]x = f(mat(x,y()))
                      [0 0 0]     [0 0 0]                 
   
                  [1]    [1]              
   chk(no(c())) = [0] >= [0] = active(c())
                  [1]    [1]              
   
                   [1 0 0]    [0]    [0]          
   mat(f(x),c()) = [0 0 0]x + [1] >= [1] = no(c())
                   [0 0 0]    [1]    [1]          
   
                  [1 0 0]    [1]    [1 0 0]    [1]               
   f(active(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = active(f(x))
                  [0 1 0]    [0]    [0 0 0]    [0]               
   
              [1 0 0]    [0]    [1 0 0]    [0]           
   f(no(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = no(f(x))
              [0 1 1]    [1]    [0 1 1]    [1]           
   
                [1 1 0]    [1]    [1 0 0]    [1]             
   f(mark(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = mark(f(x))
                [0 1 0]    [0]    [0 0 0]    [0]             
   
                 [1 1 0]    [1]    [1 1 0]                                                     
   tp(mark(x)) = [0 0 0]x + [0] >= [0 0 0]x = tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X())))))))))),x)))
                 [0 0 0]    [0]    [0 0 0]                                                     
  problem:
   active(f(x)) -> mark(f(f(x)))
   mat(f(x),f(y())) -> f(mat(x,y()))
   chk(no(c())) -> active(c())
   mat(f(x),c()) -> no(c())
   f(active(x)) -> active(f(x))
   f(no(x)) -> no(f(x))
   f(mark(x)) -> mark(f(x))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                [1 0 0]     [1]
    [chk](x0) = [0 0 0]x0 + [1]
                [0 0 0]     [0],
    
          [0]
    [c] = [0]
          [0],
    
                   [1 0 0]     [1]
    [active](x0) = [0 0 0]x0 + [0]
                   [0 0 0]     [0],
    
               [1 0 0]  
    [no](x0) = [0 0 0]x0
               [0 0 0]  ,
    
              [1 0 0]  
    [f](x0) = [0 0 0]x0
              [0 0 0]  ,
    
          [0]
    [y] = [0]
          [0],
    
                    [1 0 0]     [1 0 0]  
    [mat](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                    [0 0 0]     [0 0 0]  ,
    
                 [1 1 0]  
    [mark](x0) = [0 0 0]x0
                 [0 0 0]  
   orientation:
                   [1 0 0]    [1]    [1 0 0]                 
    active(f(x)) = [0 0 0]x + [0] >= [0 0 0]x = mark(f(f(x)))
                   [0 0 0]    [0]    [0 0 0]                 
    
                       [1 0 0]     [1 0 0]                 
    mat(f(x),f(y())) = [0 0 0]x >= [0 0 0]x = f(mat(x,y()))
                       [0 0 0]     [0 0 0]                 
    
                   [1]    [1]              
    chk(no(c())) = [1] >= [0] = active(c())
                   [0]    [0]              
    
                    [1 0 0]     [0]          
    mat(f(x),c()) = [0 0 0]x >= [0] = no(c())
                    [0 0 0]     [0]          
    
                   [1 0 0]    [1]    [1 0 0]    [1]               
    f(active(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = active(f(x))
                   [0 0 0]    [0]    [0 0 0]    [0]               
    
               [1 0 0]     [1 0 0]            
    f(no(x)) = [0 0 0]x >= [0 0 0]x = no(f(x))
               [0 0 0]     [0 0 0]            
    
                 [1 1 0]     [1 0 0]              
    f(mark(x)) = [0 0 0]x >= [0 0 0]x = mark(f(x))
                 [0 0 0]     [0 0 0]              
   problem:
    mat(f(x),f(y())) -> f(mat(x,y()))
    chk(no(c())) -> active(c())
    mat(f(x),c()) -> no(c())
    f(active(x)) -> active(f(x))
    f(no(x)) -> no(f(x))
    f(mark(x)) -> mark(f(x))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                 [1 0 0]     [1]
     [chk](x0) = [0 1 0]x0 + [0]
                 [0 0 0]     [0],
     
           [0]
     [c] = [0]
           [0],
     
                    [1 1 0]     [0]
     [active](x0) = [0 0 1]x0 + [1]
                    [0 1 0]     [0],
     
                [1 1 0]     [0]
     [no](x0) = [0 0 1]x0 + [1]
                [0 1 0]     [0],
     
               [1 1 0]     [0]
     [f](x0) = [0 0 1]x0 + [1]
               [0 1 0]     [0],
     
           [0]
     [y] = [0]
           [0],
     
                     [1 0 0]     [1 1 0]     [0]
     [mat](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1]
                     [0 0 0]     [0 1 0]     [0],
     
                       [1]
     [mark](x0) = x0 + [0]
                       [0]
    orientation:
                        [1 1 0]    [1]    [1 0 0]    [1]                
     mat(f(x),f(y())) = [0 0 0]x + [1] >= [0 0 0]x + [1] = f(mat(x,y()))
                        [0 0 0]    [1]    [0 0 0]    [1]                
     
                    [1]    [0]              
     chk(no(c())) = [1] >= [1] = active(c())
                    [0]    [0]              
     
                     [1 1 0]    [0]    [0]          
     mat(f(x),c()) = [0 0 0]x + [1] >= [1] = no(c())
                     [0 0 0]    [0]    [0]          
     
                    [1 1 1]    [1]    [1 1 1]    [1]               
     f(active(x)) = [0 1 0]x + [1] >= [0 1 0]x + [1] = active(f(x))
                    [0 0 1]    [1]    [0 0 1]    [1]               
     
                [1 1 1]    [1]    [1 1 1]    [1]           
     f(no(x)) = [0 1 0]x + [1] >= [0 1 0]x + [1] = no(f(x))
                [0 0 1]    [1]    [0 0 1]    [1]           
     
                  [1 1 0]    [1]    [1 1 0]    [1]             
     f(mark(x)) = [0 0 1]x + [1] >= [0 0 1]x + [1] = mark(f(x))
                  [0 1 0]    [0]    [0 1 0]    [0]             
    problem:
     mat(f(x),f(y())) -> f(mat(x,y()))
     mat(f(x),c()) -> no(c())
     f(active(x)) -> active(f(x))
     f(no(x)) -> no(f(x))
     f(mark(x)) -> mark(f(x))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [c] = 0,
      
      [active](x0) = 4x0 + 3,
      
      [no](x0) = 5x0 + 4,
      
      [f](x0) = 4x0 + 3,
      
      [y] = 1,
      
      [mat](x0, x1) = 4x0 + x1 + 4,
      
      [mark](x0) = 3x0 + 2
     orientation:
      mat(f(x),f(y())) = 16x + 23 >= 16x + 23 = f(mat(x,y()))
      
      mat(f(x),c()) = 16x + 16 >= 4 = no(c())
      
      f(active(x)) = 16x + 15 >= 16x + 15 = active(f(x))
      
      f(no(x)) = 20x + 19 >= 20x + 19 = no(f(x))
      
      f(mark(x)) = 12x + 11 >= 12x + 11 = mark(f(x))
     problem:
      mat(f(x),f(y())) -> f(mat(x,y()))
      f(active(x)) -> active(f(x))
      f(no(x)) -> no(f(x))
      f(mark(x)) -> mark(f(x))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [active](x0) = x0 + 2,
       
       [no](x0) = x0 + 2,
       
       [f](x0) = x0 + 1,
       
       [y] = 3,
       
       [mat](x0, x1) = x0 + 3x1 + 7,
       
       [mark](x0) = x0
      orientation:
       mat(f(x),f(y())) = x + 20 >= x + 17 = f(mat(x,y()))
       
       f(active(x)) = x + 3 >= x + 3 = active(f(x))
       
       f(no(x)) = x + 3 >= x + 3 = no(f(x))
       
       f(mark(x)) = x + 1 >= x + 1 = mark(f(x))
      problem:
       f(active(x)) -> active(f(x))
       f(no(x)) -> no(f(x))
       f(mark(x)) -> mark(f(x))
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [active](x0) = x0 + 1,
        
        [no](x0) = x0,
        
        [f](x0) = 4x0 + 4,
        
        [mark](x0) = 4x0 + 4
       orientation:
        f(active(x)) = 4x + 8 >= 4x + 5 = active(f(x))
        
        f(no(x)) = 4x + 4 >= 4x + 4 = no(f(x))
        
        f(mark(x)) = 16x + 20 >= 16x + 20 = mark(f(x))
       problem:
        f(no(x)) -> no(f(x))
        f(mark(x)) -> mark(f(x))
       Matrix Interpretation Processor: dim=3
        
        interpretation:
                    [1 0 1]     [0]
         [no](x0) = [0 1 0]x0 + [1]
                    [0 1 0]     [1],
         
                   [1 0 1]  
         [f](x0) = [0 1 0]x0
                   [0 1 0]  ,
         
                      [1 0 1]     [1]
         [mark](x0) = [0 1 0]x0 + [0]
                      [0 1 0]     [0]
        orientation:
                    [1 1 1]    [1]    [1 1 1]    [0]           
         f(no(x)) = [0 1 0]x + [1] >= [0 1 0]x + [1] = no(f(x))
                    [0 1 0]    [1]    [0 1 0]    [1]           
         
                      [1 1 1]    [1]    [1 1 1]    [1]             
         f(mark(x)) = [0 1 0]x + [0] >= [0 1 0]x + [0] = mark(f(x))
                      [0 1 0]    [0]    [0 1 0]    [0]             
        problem:
         f(mark(x)) -> mark(f(x))
        Matrix Interpretation Processor: dim=3
         
         interpretation:
                    [1 1 0]  
          [f](x0) = [0 1 0]x0
                    [0 0 0]  ,
          
                       [1 0 0]     [0]
          [mark](x0) = [0 1 0]x0 + [1]
                       [0 0 0]     [0]
         orientation:
                       [1 1 0]    [1]    [1 1 0]    [0]             
          f(mark(x)) = [0 1 0]x + [1] >= [0 1 0]x + [1] = mark(f(x))
                       [0 0 0]    [0]    [0 0 0]    [0]             
         problem:
          
         Qed