YES

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

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