YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(a(x1))) -> b#(b(a(x1)))
   b#(b(b(x1))) -> a#(x1)
   b#(b(b(x1))) -> b#(a(x1))
   b#(b(x1)) -> a#(x1)
   b#(b(x1)) -> a#(a(x1))
   b#(b(x1)) -> a#(a(a(x1)))
  TRS:
   a(b(a(x1))) -> b(b(a(x1)))
   b(b(b(x1))) -> b(a(x1))
   b(b(x1)) -> a(a(a(x1)))
  TDG Processor:
   DPs:
    a#(b(a(x1))) -> b#(b(a(x1)))
    b#(b(b(x1))) -> a#(x1)
    b#(b(b(x1))) -> b#(a(x1))
    b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> a#(a(x1))
    b#(b(x1)) -> a#(a(a(x1)))
   TRS:
    a(b(a(x1))) -> b(b(a(x1)))
    b(b(b(x1))) -> b(a(x1))
    b(b(x1)) -> a(a(a(x1)))
   graph:
    b#(b(b(x1))) -> b#(a(x1)) -> b#(b(x1)) -> a#(a(a(x1)))
    b#(b(b(x1))) -> b#(a(x1)) -> b#(b(x1)) -> a#(a(x1))
    b#(b(b(x1))) -> b#(a(x1)) -> b#(b(x1)) -> a#(x1)
    b#(b(b(x1))) -> b#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1))
    b#(b(b(x1))) -> b#(a(x1)) -> b#(b(b(x1))) -> a#(x1)
    b#(b(b(x1))) -> a#(x1) -> a#(b(a(x1))) -> b#(b(a(x1)))
    b#(b(x1)) -> a#(a(a(x1))) -> a#(b(a(x1))) -> b#(b(a(x1)))
    b#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1))) -> b#(b(a(x1)))
    b#(b(x1)) -> a#(x1) -> a#(b(a(x1))) -> b#(b(a(x1)))
    a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(a(a(x1)))
    a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(a(x1))
    a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(x1)
    a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(x1))) -> b#(a(x1))
    a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(x1))) -> a#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    usable rules:
     a(b(a(x1))) -> b(b(a(x1)))
     b(b(b(x1))) -> b(a(x1))
     b(b(x1)) -> a(a(a(x1)))
    interpretation:
               [0 2]     [2]
     [b](x0) = [0 0]x0 + [0],
     
     [b#](x0) = [-& 0 ]x0 + [0],
     
               [0  -&]     [0 ]
     [a](x0) = [0  0 ]x0 + [-&],
     
     [a#](x0) = [-& 0 ]x0 + [0]
    orientation:
     a#(b(a(x1))) = [0 0]x1 + [0] >= [0 0]x1 + [0] = b#(b(a(x1)))
     
     b#(b(b(x1))) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a#(x1)
     
     b#(b(b(x1))) = [0 2]x1 + [2] >= [0 0]x1 + [0] = b#(a(x1))
     
     b#(b(x1)) = [0 0]x1 + [0] >= [-& 0 ]x1 + [0] = a#(x1)
     
     b#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a#(a(x1))
     
     b#(b(x1)) = [0 0]x1 + [0] >= [0 0]x1 + [0] = a#(a(a(x1)))
     
                   [2 2]     [2]    [2 2]     [2]              
     a(b(a(x1))) = [2 2]x1 + [2] >= [2 2]x1 + [2] = b(b(a(x1)))
     
                   [2 4]     [4]    [2 2]     [2]           
     b(b(b(x1))) = [2 2]x1 + [2] >= [0 0]x1 + [0] = b(a(x1))
     
                [2 2]     [2]    [0  -&]     [0]              
     b(b(x1)) = [0 2]x1 + [2] >= [0  0 ]x1 + [0] = a(a(a(x1)))
    problem:
     DPs:
      a#(b(a(x1))) -> b#(b(a(x1)))
      b#(b(b(x1))) -> b#(a(x1))
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> a#(a(x1))
      b#(b(x1)) -> a#(a(a(x1)))
     TRS:
      a(b(a(x1))) -> b(b(a(x1)))
      b(b(b(x1))) -> b(a(x1))
      b(b(x1)) -> a(a(a(x1)))
    Restore Modifier:
     DPs:
      a#(b(a(x1))) -> b#(b(a(x1)))
      b#(b(b(x1))) -> b#(a(x1))
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> a#(a(x1))
      b#(b(x1)) -> a#(a(a(x1)))
     TRS:
      a(b(a(x1))) -> b(b(a(x1)))
      b(b(b(x1))) -> b(a(x1))
      b(b(x1)) -> a(a(a(x1)))
     EDG Processor:
      DPs:
       a#(b(a(x1))) -> b#(b(a(x1)))
       b#(b(b(x1))) -> b#(a(x1))
       b#(b(x1)) -> a#(x1)
       b#(b(x1)) -> a#(a(x1))
       b#(b(x1)) -> a#(a(a(x1)))
      TRS:
       a(b(a(x1))) -> b(b(a(x1)))
       b(b(b(x1))) -> b(a(x1))
       b(b(x1)) -> a(a(a(x1)))
      graph:
       b#(b(b(x1))) -> b#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1))
       b#(b(b(x1))) -> b#(a(x1)) -> b#(b(x1)) -> a#(x1)
       b#(b(b(x1))) -> b#(a(x1)) -> b#(b(x1)) -> a#(a(x1))
       b#(b(b(x1))) -> b#(a(x1)) -> b#(b(x1)) -> a#(a(a(x1)))
       b#(b(x1)) -> a#(a(a(x1))) -> a#(b(a(x1))) -> b#(b(a(x1)))
       b#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1))) -> b#(b(a(x1)))
       b#(b(x1)) -> a#(x1) -> a#(b(a(x1))) -> b#(b(a(x1)))
       a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(b(x1))) -> b#(a(x1))
       a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(x1)
       a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(a(x1))
       a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(a(a(x1)))
      Arctic Interpretation Processor:
       dimension: 2
       usable rules:
        a(b(a(x1))) -> b(b(a(x1)))
        b(b(b(x1))) -> b(a(x1))
        b(b(x1)) -> a(a(a(x1)))
       interpretation:
                  [-& 0 ]     [0]
        [b](x0) = [2  0 ]x0 + [0],
        
        [b#](x0) = [0 0]x0 + [0],
        
                  [0 0]     [0]
        [a](x0) = [0 0]x0 + [0],
        
        [a#](x0) = [0 0]x0 + [0]
       orientation:
        a#(b(a(x1))) = [2 2]x1 + [2] >= [2 2]x1 + [2] = b#(b(a(x1)))
        
        b#(b(b(x1))) = [2 2]x1 + [2] >= [0 0]x1 + [0] = b#(a(x1))
        
        b#(b(x1)) = [2 0]x1 + [0] >= [0 0]x1 + [0] = a#(x1)
        
        b#(b(x1)) = [2 0]x1 + [0] >= [0 0]x1 + [0] = a#(a(x1))
        
        b#(b(x1)) = [2 0]x1 + [0] >= [0 0]x1 + [0] = a#(a(a(x1)))
        
                      [2 2]     [2]    [2 2]     [2]              
        a(b(a(x1))) = [2 2]x1 + [2] >= [2 2]x1 + [2] = b(b(a(x1)))
        
                      [2 2]     [2]    [0 0]     [0]           
        b(b(b(x1))) = [4 2]x1 + [2] >= [2 2]x1 + [2] = b(a(x1))
        
                   [2 0]     [0]    [0 0]     [0]              
        b(b(x1)) = [2 2]x1 + [2] >= [0 0]x1 + [0] = a(a(a(x1)))
       problem:
        DPs:
         a#(b(a(x1))) -> b#(b(a(x1)))
         b#(b(x1)) -> a#(x1)
         b#(b(x1)) -> a#(a(x1))
         b#(b(x1)) -> a#(a(a(x1)))
        TRS:
         a(b(a(x1))) -> b(b(a(x1)))
         b(b(b(x1))) -> b(a(x1))
         b(b(x1)) -> a(a(a(x1)))
       Restore Modifier:
        DPs:
         a#(b(a(x1))) -> b#(b(a(x1)))
         b#(b(x1)) -> a#(x1)
         b#(b(x1)) -> a#(a(x1))
         b#(b(x1)) -> a#(a(a(x1)))
        TRS:
         a(b(a(x1))) -> b(b(a(x1)))
         b(b(b(x1))) -> b(a(x1))
         b(b(x1)) -> a(a(a(x1)))
        EDG Processor:
         DPs:
          a#(b(a(x1))) -> b#(b(a(x1)))
          b#(b(x1)) -> a#(x1)
          b#(b(x1)) -> a#(a(x1))
          b#(b(x1)) -> a#(a(a(x1)))
         TRS:
          a(b(a(x1))) -> b(b(a(x1)))
          b(b(b(x1))) -> b(a(x1))
          b(b(x1)) -> a(a(a(x1)))
         graph:
          b#(b(x1)) -> a#(a(a(x1))) -> a#(b(a(x1))) -> b#(b(a(x1)))
          b#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1))) -> b#(b(a(x1)))
          b#(b(x1)) -> a#(x1) -> a#(b(a(x1))) -> b#(b(a(x1)))
          a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(a(a(x1)))
          a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(a(x1))
          a#(b(a(x1))) -> b#(b(a(x1))) -> b#(b(x1)) -> a#(x1)
         Arctic Interpretation Processor:
          dimension: 2
          usable rules:
           a(b(a(x1))) -> b(b(a(x1)))
           b(b(b(x1))) -> b(a(x1))
           b(b(x1)) -> a(a(a(x1)))
          interpretation:
                     [0 1]     [0]
           [b](x0) = [0 0]x0 + [3],
           
           [b#](x0) = [-& 0 ]x0 + [0],
           
                     [0  -&]     [-&]
           [a](x0) = [0  -&]x0 + [3 ],
           
           [a#](x0) = [0 0]x0 + [0]
          orientation:
           a#(b(a(x1))) = [1  -&]x1 + [4] >= [0  -&]x1 + [3] = b#(b(a(x1)))
           
           b#(b(x1)) = [0 0]x1 + [3] >= [0 0]x1 + [0] = a#(x1)
           
           b#(b(x1)) = [0 0]x1 + [3] >= [0  -&]x1 + [3] = a#(a(x1))
           
           b#(b(x1)) = [0 0]x1 + [3] >= [0  -&]x1 + [3] = a#(a(a(x1)))
           
                         [1  -&]     [4]    [1  -&]     [4]              
           a(b(a(x1))) = [1  -&]x1 + [4] >= [1  -&]x1 + [4] = b(b(a(x1)))
           
                         [1 2]     [4]    [1  -&]     [4]           
           b(b(b(x1))) = [1 1]x1 + [4] >= [0  -&]x1 + [3] = b(a(x1))
           
                      [1 1]     [4]    [0  -&]     [-&]              
           b(b(x1)) = [0 1]x1 + [3] >= [0  -&]x1 + [3 ] = a(a(a(x1)))
          problem:
           DPs:
            b#(b(x1)) -> a#(x1)
            b#(b(x1)) -> a#(a(x1))
            b#(b(x1)) -> a#(a(a(x1)))
           TRS:
            a(b(a(x1))) -> b(b(a(x1)))
            b(b(b(x1))) -> b(a(x1))
            b(b(x1)) -> a(a(a(x1)))
          Restore Modifier:
           DPs:
            b#(b(x1)) -> a#(x1)
            b#(b(x1)) -> a#(a(x1))
            b#(b(x1)) -> a#(a(a(x1)))
           TRS:
            a(b(a(x1))) -> b(b(a(x1)))
            b(b(b(x1))) -> b(a(x1))
            b(b(x1)) -> a(a(a(x1)))
           EDG Processor:
            DPs:
             b#(b(x1)) -> a#(x1)
             b#(b(x1)) -> a#(a(x1))
             b#(b(x1)) -> a#(a(a(x1)))
            TRS:
             a(b(a(x1))) -> b(b(a(x1)))
             b(b(b(x1))) -> b(a(x1))
             b(b(x1)) -> a(a(a(x1)))
            graph:
             
            SCC Processor:
             #sccs: 0
             #rules: 0
             #arcs: 0/9