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
    usable rules:
     a(x1) -> x1
     a(a(x1)) -> b(x1)
     b(a(b(x1))) -> a(a(b(b(b(x1)))))
    interpretation:
     [b#](x0) = [0 0]x0 + [0],
     
     [a#](x0) = [0 3]x0 + [0],
     
               [0 0]     [-&]
     [b](x0) = [0 0]x0 + [1 ],
     
               [0 3]     [1]
     [a](x0) = [0 0]x0 + [0]
    orientation:
     a#(a(x1)) = [3 3]x1 + [3] >= [0 0]x1 + [0] = b#(x1)
     
     b#(a(b(x1))) = [3 3]x1 + [4] >= [0 0]x1 + [1] = b#(b(x1))
     
     b#(a(b(x1))) = [3 3]x1 + [4] >= [0 0]x1 + [1] = b#(b(b(x1)))
     
     b#(a(b(x1))) = [3 3]x1 + [4] >= [3 3]x1 + [4] = a#(b(b(b(x1))))
     
     b#(a(b(x1))) = [3 3]x1 + [4] >= [3 3]x1 + [4] = a#(a(b(b(b(x1)))))
     
             [0 3]     [1]           
     a(x1) = [0 0]x1 + [0] >= x1 = x1
     
                [3 3]     [3]    [0 0]     [-&]        
     a(a(x1)) = [0 3]x1 + [1] >= [0 0]x1 + [1 ] = b(x1)
     
                   [3 3]     [4]    [3 3]     [4]                    
     b(a(b(x1))) = [3 3]x1 + [4] >= [3 3]x1 + [4] = a(a(b(b(b(x1)))))
    problem:
     DPs:
      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)))))
    Restore Modifier:
     DPs:
      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)))))
     EDG Processor:
      DPs:
       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:
       
      Qed