YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(a(x1))) -> b#(x1)
   b#(b(x1)) -> a#(b(x1))
   b#(b(x1)) -> b#(a(b(x1)))
   b#(b(x1)) -> a#(x1)
   b#(b(x1)) -> a#(a(x1))
  TRS:
   a(a(a(x1))) -> b(x1)
   b(b(x1)) -> b(a(b(x1)))
   b(b(x1)) -> a(a(x1))
  TDG Processor:
   DPs:
    a#(a(a(x1))) -> b#(x1)
    b#(b(x1)) -> a#(b(x1))
    b#(b(x1)) -> b#(a(b(x1)))
    b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> a#(a(x1))
   TRS:
    a(a(a(x1))) -> b(x1)
    b(b(x1)) -> b(a(b(x1)))
    b(b(x1)) -> a(a(x1))
   graph:
    b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(a(x1))
    b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1)))
    b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(b(x1))
    b#(b(x1)) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1)
    b#(b(x1)) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1)
    b#(b(x1)) -> a#(x1) -> a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(a(x1))
    a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(x1)
    a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> b#(a(b(x1)))
    a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(b(x1))
   Arctic Interpretation Processor:
    dimension: 2
    usable rules:
     a(a(a(x1))) -> b(x1)
     b(b(x1)) -> b(a(b(x1)))
     b(b(x1)) -> a(a(x1))
    interpretation:
               [0 1]     [0]
     [b](x0) = [0 1]x0 + [0],
     
     [b#](x0) = [-& 0 ]x0 + [0],
     
               [0  1 ]     [0]
     [a](x0) = [0  -&]x0 + [0],
     
     [a#](x0) = [0 0]x0 + [0]
    orientation:
     a#(a(a(x1))) = [1 1]x1 + [1] >= [-& 0 ]x1 + [0] = b#(x1)
     
     b#(b(x1)) = [0 1]x1 + [0] >= [0 1]x1 + [0] = a#(b(x1))
     
     b#(b(x1)) = [0 1]x1 + [0] >= [0 1]x1 + [0] = b#(a(b(x1)))
     
     b#(b(x1)) = [0 1]x1 + [0] >= [0 0]x1 + [0] = a#(x1)
     
     b#(b(x1)) = [0 1]x1 + [0] >= [0 1]x1 + [0] = a#(a(x1))
     
                   [1 2]     [1]    [0 1]     [0]        
     a(a(a(x1))) = [1 1]x1 + [1] >= [0 1]x1 + [0] = b(x1)
     
                [1 2]     [1]    [1 2]     [1]              
     b(b(x1)) = [1 2]x1 + [1] >= [1 2]x1 + [1] = b(a(b(x1)))
     
                [1 2]     [1]    [1 1]     [1]           
     b(b(x1)) = [1 2]x1 + [1] >= [0 1]x1 + [0] = a(a(x1))
    problem:
     DPs:
      b#(b(x1)) -> a#(b(x1))
      b#(b(x1)) -> b#(a(b(x1)))
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> a#(a(x1))
     TRS:
      a(a(a(x1))) -> b(x1)
      b(b(x1)) -> b(a(b(x1)))
      b(b(x1)) -> a(a(x1))
    Restore Modifier:
     DPs:
      b#(b(x1)) -> a#(b(x1))
      b#(b(x1)) -> b#(a(b(x1)))
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> a#(a(x1))
     TRS:
      a(a(a(x1))) -> b(x1)
      b(b(x1)) -> b(a(b(x1)))
      b(b(x1)) -> a(a(x1))
     EDG Processor:
      DPs:
       b#(b(x1)) -> a#(b(x1))
       b#(b(x1)) -> b#(a(b(x1)))
       b#(b(x1)) -> a#(x1)
       b#(b(x1)) -> a#(a(x1))
      TRS:
       a(a(a(x1))) -> b(x1)
       b(b(x1)) -> b(a(b(x1)))
       b(b(x1)) -> a(a(x1))
      graph:
       b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(b(x1))
       b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1)))
       b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(x1)
       b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(a(x1))
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 4/16
       DPs:
        b#(b(x1)) -> b#(a(b(x1)))
       TRS:
        a(a(a(x1))) -> b(x1)
        b(b(x1)) -> b(a(b(x1)))
        b(b(x1)) -> a(a(x1))
       Arctic Interpretation Processor:
        dimension: 2
        usable rules:
         a(a(a(x1))) -> b(x1)
         b(b(x1)) -> b(a(b(x1)))
         b(b(x1)) -> a(a(x1))
        interpretation:
                   [3  0 ]     [1]
         [b](x0) = [2  -&]x0 + [0],
         
         [b#](x0) = [0  -&]x0 + [0],
         
                   [-& 0 ]     [0]
         [a](x0) = [2  1 ]x0 + [0]
        orientation:
         b#(b(x1)) = [3 0]x1 + [1] >= [2  -&]x1 + [0] = b#(a(b(x1)))
         
                       [3 2]     [2]    [3  0 ]     [1]        
         a(a(a(x1))) = [4 3]x1 + [3] >= [2  -&]x1 + [0] = b(x1)
         
                    [6 3]     [4]    [5  2 ]     [3]              
         b(b(x1)) = [5 2]x1 + [3] >= [4  -&]x1 + [2] = b(a(b(x1)))
         
                    [6 3]     [4]    [2 1]     [0]           
         b(b(x1)) = [5 2]x1 + [3] >= [3 2]x1 + [2] = a(a(x1))
        problem:
         DPs:
          
         TRS:
          a(a(a(x1))) -> b(x1)
          b(b(x1)) -> b(a(b(x1)))
          b(b(x1)) -> a(a(x1))
        Qed