YES

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

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