YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   b#(a(b(x1))) -> b#(b(x1))
   b#(a(b(x1))) -> b#(b(b(x1)))
   b#(a(b(x1))) -> a#(b(b(b(x1))))
   b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
  TRS:
   a(x1) -> x1
   a(a(x1)) -> b(x1)
   b(a(b(x1))) -> a(a(b(b(b(x1)))))
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    b#(a(b(x1))) -> b#(b(x1))
    b#(a(b(x1))) -> b#(b(b(x1)))
    b#(a(b(x1))) -> a#(b(b(b(x1))))
    b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
   TRS:
    a(x1) -> x1
    a(a(x1)) -> b(x1)
    b(a(b(x1))) -> a(a(b(b(b(x1)))))
   graph:
    b#(a(b(x1))) -> b#(b(b(x1))) -> b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
    b#(a(b(x1))) -> b#(b(b(x1))) -> b#(a(b(x1))) -> a#(b(b(b(x1))))
    b#(a(b(x1))) -> b#(b(b(x1))) -> b#(a(b(x1))) -> b#(b(b(x1)))
    b#(a(b(x1))) -> b#(b(b(x1))) -> b#(a(b(x1))) -> b#(b(x1))
    b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
    b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> a#(b(b(b(x1))))
    b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(b(b(x1)))
    b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1))
    b#(a(b(x1))) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(x1)
    b#(a(b(x1))) -> a#(a(b(b(b(x1))))) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
    a#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(b(b(b(x1))))
    a#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> b#(b(b(x1)))
    a#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> b#(b(x1))
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
     [b#](x0) = [0 0]x0 + [0],
     
     [a#](x0) = [0 0]x0 + [0],
     
               [0 0]     [0]
     [b](x0) = [0 0]x0 + [0],
     
               [0 0]     [0]
     [a](x0) = [1 0]x0 + [0]
    orientation:
     a#(a(x1)) = [1 0]x1 + [0] >= [0 0]x1 + [0] = b#(x1)
     
     b#(a(b(x1))) = [1 1]x1 + [1] >= [0 0]x1 + [0] = b#(b(x1))
     
     b#(a(b(x1))) = [1 1]x1 + [1] >= [0 0]x1 + [0] = b#(b(b(x1)))
     
     b#(a(b(x1))) = [1 1]x1 + [1] >= [0 0]x1 + [0] = a#(b(b(b(x1))))
     
     b#(a(b(x1))) = [1 1]x1 + [1] >= [1 1]x1 + [1] = a#(a(b(b(b(x1)))))
     
             [0 0]     [0]           
     a(x1) = [1 0]x1 + [0] >= x1 = x1
     
                [1 0]     [0]    [0 0]     [0]        
     a(a(x1)) = [1 1]x1 + [1] >= [0 0]x1 + [0] = b(x1)
     
                   [1 1]     [1]    [1 1]     [1]                    
     b(a(b(x1))) = [1 1]x1 + [1] >= [1 1]x1 + [1] = a(a(b(b(b(x1)))))
    problem:
     DPs:
      a#(a(x1)) -> b#(x1)
      b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
     TRS:
      a(x1) -> x1
      a(a(x1)) -> b(x1)
      b(a(b(x1))) -> a(a(b(b(b(x1)))))
    EDG Processor:
     DPs:
      a#(a(x1)) -> b#(x1)
      b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
     TRS:
      a(x1) -> x1
      a(a(x1)) -> b(x1)
      b(a(b(x1))) -> a(a(b(b(b(x1)))))
     graph:
      b#(a(b(x1))) -> a#(a(b(b(b(x1))))) -> a#(a(x1)) -> b#(x1)
      a#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [b#](x0) = [2 0]x0,
       
       [a#](x0) = [2 3]x0 + [0],
       
                 [0 0]     [0]
       [b](x0) = [0 0]x0 + [2],
       
                 [0 2]     [2 ]
       [a](x0) = [0 0]x0 + [-&]
      orientation:
       a#(a(x1)) = [3 4]x1 + [4] >= [2 0]x1 = b#(x1)
       
       b#(a(b(x1))) = [4 4]x1 + [6] >= [4 4]x1 + [6] = a#(a(b(b(b(x1)))))
       
               [0 2]     [2 ]           
       a(x1) = [0 0]x1 + [-&] >= x1 = x1
       
                  [2 2]     [2]    [0 0]     [0]        
       a(a(x1)) = [0 2]x1 + [2] >= [0 0]x1 + [2] = b(x1)
       
                     [2 2]     [4]    [2 2]     [4]                    
       b(a(b(x1))) = [2 2]x1 + [4] >= [2 2]x1 + [4] = a(a(b(b(b(x1)))))
      problem:
       DPs:
        b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
       TRS:
        a(x1) -> x1
        a(a(x1)) -> b(x1)
        b(a(b(x1))) -> a(a(b(b(b(x1)))))
      EDG Processor:
       DPs:
        b#(a(b(x1))) -> a#(a(b(b(b(x1)))))
       TRS:
        a(x1) -> x1
        a(a(x1)) -> b(x1)
        b(a(b(x1))) -> a(a(b(b(b(x1)))))
       graph:
        
       Qed