YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(a(a(b(b(x1)))))) -> b#(a(a(b(b(x1)))))
   a#(a(a(a(b(b(x1)))))) -> b#(b(a(a(b(b(x1))))))
   b#(b(a(a(x1)))) -> b#(x1)
   b#(b(a(a(x1)))) -> b#(b(x1))
   b#(b(a(a(x1)))) -> b#(b(b(x1)))
   b#(b(a(a(x1)))) -> b#(b(b(b(x1))))
   b#(b(a(a(x1)))) -> a#(b(b(b(b(x1)))))
   b#(b(a(a(x1)))) -> a#(a(b(b(b(b(x1))))))
   b#(b(c(c(a(a(x1)))))) -> b#(x1)
   b#(b(c(c(a(a(x1)))))) -> b#(b(x1))
   b#(b(c(c(a(a(x1)))))) -> a#(b(b(x1)))
   b#(b(c(c(a(a(x1)))))) -> a#(a(b(b(x1))))
   b#(b(c(c(a(a(x1)))))) -> a#(a(a(b(b(x1)))))
   b#(b(c(c(a(a(x1)))))) -> a#(a(a(a(b(b(x1))))))
  TRS:
   a(a(a(a(b(b(x1)))))) -> b(b(a(a(b(b(x1))))))
   b(b(a(a(x1)))) -> a(a(b(b(b(b(x1))))))
   b(b(c(c(a(a(x1)))))) -> c(c(c(c(a(a(a(a(b(b(x1))))))))))
  Matrix Interpretation Processor: dim=5
   
   interpretation:
    [b#](x0) = [0 0 0 0 1]x0,
    
    [a#](x0) = [0 1 0 0 0]x0,
    
              [0 0 0 1 1]  
              [0 1 0 0 0]  
    [c](x0) = [0 0 0 0 0]x0
              [0 0 1 0 0]  
              [0 0 1 0 0]  ,
    
              [1 0 0 0 0]     [0]
              [0 1 1 0 0]     [0]
    [a](x0) = [1 0 0 1 1]x0 + [1]
              [0 0 0 0 0]     [0]
              [0 0 0 0 0]     [0],
    
              [0 0 0 0 0]  
              [0 0 0 0 1]  
    [b](x0) = [1 0 1 0 0]x0
              [0 0 0 0 0]  
              [1 1 0 0 0]  
   orientation:
    a#(a(a(a(b(b(x1)))))) = [2 1 1 0 1]x1 + [2] >= [0] = b#(a(a(b(b(x1)))))
    
    a#(a(a(a(b(b(x1)))))) = [2 1 1 0 1]x1 + [2] >= [2 1 1 0 1]x1 + [1] = b#(b(a(a(b(b(x1))))))
    
    b#(b(a(a(x1)))) = [2 1 1 1 1]x1 + [1] >= [0 0 0 0 1]x1 = b#(x1)
    
    b#(b(a(a(x1)))) = [2 1 1 1 1]x1 + [1] >= [1 1 0 0 0]x1 = b#(b(x1))
    
    b#(b(a(a(x1)))) = [2 1 1 1 1]x1 + [1] >= [0 0 0 0 1]x1 = b#(b(b(x1)))
    
    b#(b(a(a(x1)))) = [2 1 1 1 1]x1 + [1] >= [1 1 0 0 0]x1 = b#(b(b(b(x1))))
    
    b#(b(a(a(x1)))) = [2 1 1 1 1]x1 + [1] >= [1 1 0 0 0]x1 = a#(b(b(b(b(x1)))))
    
    b#(b(a(a(x1)))) = [2 1 1 1 1]x1 + [1] >= [2 1 1 0 0]x1 = a#(a(b(b(b(b(x1))))))
    
    b#(b(c(c(a(a(x1)))))) = [3 1 1 1 1]x1 + [3] >= [0 0 0 0 1]x1 = b#(x1)
    
    b#(b(c(c(a(a(x1)))))) = [3 1 1 1 1]x1 + [3] >= [1 1 0 0 0]x1 = b#(b(x1))
    
    b#(b(c(c(a(a(x1)))))) = [3 1 1 1 1]x1 + [3] >= [1 1 0 0 0]x1 = a#(b(b(x1)))
    
    b#(b(c(c(a(a(x1)))))) = [3 1 1 1 1]x1 + [3] >= [2 1 1 0 0]x1 = a#(a(b(b(x1))))
    
    b#(b(c(c(a(a(x1)))))) = [3 1 1 1 1]x1 + [3] >= [2 1 1 0 1]x1 + [1] = a#(a(a(b(b(x1)))))
    
    b#(b(c(c(a(a(x1)))))) = [3 1 1 1 1]x1 + [3] >= [2 1 1 0 1]x1 + [2] = a#(a(a(a(b(b(x1))))))
    
                           [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                       
                           [2 1 1 0 1]     [3]    [2 1 1 0 1]     [1]                       
    a(a(a(a(b(b(x1)))))) = [0 0 0 0 0]x1 + [1] >= [0 0 0 0 0]x1 + [1] = b(b(a(a(b(b(x1))))))
                           [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                       
                           [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                       
    
                     [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                       
                     [2 1 1 1 1]     [1]    [2 1 1 0 1]     [1]                       
    b(b(a(a(x1)))) = [2 0 0 0 0]x1 + [1] >= [0 0 0 0 0]x1 + [1] = a(a(b(b(b(b(x1))))))
                     [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                       
                     [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                       
    
                           [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                                   
                           [3 1 1 1 1]     [3]    [2 1 1 0 1]     [3]                                   
    b(b(c(c(a(a(x1)))))) = [2 0 0 0 0]x1 + [2] >= [0 0 0 0 0]x1 + [0] = c(c(c(c(a(a(a(a(b(b(x1))))))))))
                           [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                                   
                           [0 0 0 0 0]     [0]    [0 0 0 0 0]     [0]                                   
   problem:
    DPs:
     
    TRS:
     a(a(a(a(b(b(x1)))))) -> b(b(a(a(b(b(x1))))))
     b(b(a(a(x1)))) -> a(a(b(b(b(b(x1))))))
     b(b(c(c(a(a(x1)))))) -> c(c(c(c(a(a(a(a(b(b(x1))))))))))
   Qed