YES(?,O(n^2))

Problem:
 a(a(x1)) -> b(x1)
 a(a(a(x1))) -> a(b(a(x1)))
 a(b(a(x1))) -> b(b(b(x1)))
 a(a(a(a(x1)))) -> a(a(b(a(a(x1)))))
 a(a(b(a(x1)))) -> a(b(b(a(b(x1)))))
 a(b(a(a(x1)))) -> b(a(b(b(a(x1)))))
 a(b(b(a(x1)))) -> b(b(b(b(b(x1)))))
 a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1)))))))
 a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1)))))))
 a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1)))))))
 a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1)))))))
 a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1)))))))
 a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1)))))))
 a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1)))))))
 a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1)))))))

Proof:
 Matrix Interpretation Processor:
  dimension: 2
  interpretation:
             [1 0]  
   [b](x0) = [0 0]x0,
   
             [1 1]     [1]
   [a](x0) = [0 1]x0 + [2]
  orientation:
              [1 2]     [4]    [1 0]          
   a(a(x1)) = [0 1]x1 + [4] >= [0 0]x1 = b(x1)
   
                 [1 3]     [9]    [1 1]     [2]              
   a(a(a(x1))) = [0 1]x1 + [6] >= [0 0]x1 + [2] = a(b(a(x1)))
   
                 [1 1]     [2]    [1 0]                
   a(b(a(x1))) = [0 0]x1 + [2] >= [0 0]x1 = b(b(b(x1)))
   
                    [1 4]     [16]    [1 2]     [8]                    
   a(a(a(a(x1)))) = [0 1]x1 + [8 ] >= [0 0]x1 + [4] = a(a(b(a(a(x1)))))
   
                    [1 1]     [5]    [1 0]     [2]                    
   a(a(b(a(x1)))) = [0 0]x1 + [4] >= [0 0]x1 + [2] = a(b(b(a(b(x1)))))
   
                    [1 2]     [5]    [1 1]     [2]                    
   a(b(a(a(x1)))) = [0 0]x1 + [2] >= [0 0]x1 + [0] = b(a(b(b(a(x1)))))
   
                    [1 1]     [2]    [1 0]                      
   a(b(b(a(x1)))) = [0 0]x1 + [2] >= [0 0]x1 = b(b(b(b(b(x1)))))
   
                       [1 5]     [25]    [1 3]     [18]                          
   a(a(a(a(a(x1))))) = [0 1]x1 + [10] >= [0 0]x1 + [6 ] = a(a(a(b(a(a(a(x1)))))))
   
                       [1 1]     [10]    [1 0]     [8]                          
   a(a(a(b(a(x1))))) = [0 0]x1 + [6 ] >= [0 0]x1 + [4] = a(a(b(b(a(a(b(x1)))))))
   
                       [1 2]     [8]    [1 1]     [4]                          
   a(a(b(a(a(x1))))) = [0 0]x1 + [4] >= [0 0]x1 + [2] = a(b(a(b(a(b(a(x1)))))))
   
                       [1 1]     [5]    [1 0]     [2]                          
   a(a(b(b(a(x1))))) = [0 0]x1 + [4] >= [0 0]x1 + [2] = a(b(b(b(a(b(b(x1)))))))
   
                       [1 3]     [10]    [1 2]     [8]                          
   a(b(a(a(a(x1))))) = [0 0]x1 + [2 ] >= [0 0]x1 + [0] = b(a(a(b(b(a(a(x1)))))))
   
                       [1 1]     [3]    [1 0]     [2]                          
   a(b(a(b(a(x1))))) = [0 0]x1 + [2] >= [0 0]x1 + [0] = b(a(b(b(b(a(b(x1)))))))
   
                       [1 2]     [5]    [1 1]     [2]                          
   a(b(b(a(a(x1))))) = [0 0]x1 + [2] >= [0 0]x1 + [0] = b(b(a(b(b(b(a(x1)))))))
   
                       [1 1]     [2]    [1 0]                            
   a(b(b(b(a(x1))))) = [0 0]x1 + [2] >= [0 0]x1 = b(b(b(b(b(b(b(x1)))))))
  problem:
   
  Qed