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:
               [0 0]     [0 ]
     [b](x0) = [0 0]x0 + [-&],
     
     [b#](x0) = [0  -&]x0,
     
               [0 1]     [3]
     [a](x0) = [0 0]x0 + [2],
     
     [a#](x0) = [0  -&]x0 + [3]
    orientation:
     a#(a(x1)) = [0 1]x1 + [3] >= [0  -&]x1 = b#(x1)
     
     b#(a(b(x1))) = [1 1]x1 + [3] >= [0 0]x1 + [0] = b#(b(x1))
     
     b#(a(b(x1))) = [1 1]x1 + [3] >= [0 0]x1 + [0] = b#(b(b(x1)))
     
     b#(a(b(x1))) = [1 1]x1 + [3] >= [0 0]x1 + [3] = a#(b(b(b(x1))))
     
     b#(a(b(x1))) = [1 1]x1 + [3] >= [1 1]x1 + [3] = a#(a(b(b(b(x1)))))
     
             [0 1]     [3]           
     a(x1) = [0 0]x1 + [2] >= x1 = x1
     
                [1 1]     [3]    [0 0]     [0 ]        
     a(a(x1)) = [0 1]x1 + [3] >= [0 0]x1 + [-&] = b(x1)
     
                   [1 1]     [3]    [1 1]     [3]                    
     b(a(b(x1))) = [1 1]x1 + [3] >= [1 1]x1 + [3] = a(a(b(b(b(x1)))))
    problem:
     DPs:
      a#(a(x1)) -> 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)))))
    Restore Modifier:
     DPs:
      a#(a(x1)) -> 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)))))
     EDG Processor:
      DPs:
       a#(a(x1)) -> 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))) -> 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#(b(b(b(x1))))
       a#(a(x1)) -> b#(x1) -> b#(a(b(x1))) -> a#(a(b(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:
                  [-& 3 ]     [0 ]
        [b](x0) = [-& 0 ]x0 + [-&],
        
        [b#](x0) = [1 0]x0 + [3],
        
                  [0 3]     [1]
        [a](x0) = [0 0]x0 + [2],
        
        [a#](x0) = [0 1]x0 + [0]
       orientation:
        a#(a(x1)) = [1 3]x1 + [3] >= [1 0]x1 + [3] = b#(x1)
        
        b#(a(b(x1))) = [-& 4 ]x1 + [3] >= [-& 3 ]x1 + [0] = a#(b(b(b(x1))))
        
        b#(a(b(x1))) = [-& 4 ]x1 + [3] >= [-& 4 ]x1 + [3] = a#(a(b(b(b(x1)))))
        
                [0 3]     [1]           
        a(x1) = [0 0]x1 + [2] >= x1 = x1
        
                   [3 3]     [5]    [-& 3 ]     [0 ]        
        a(a(x1)) = [0 3]x1 + [2] >= [-& 0 ]x1 + [-&] = b(x1)
        
                      [-& 6 ]     [5]    [-& 6 ]     [5]                    
        b(a(b(x1))) = [-& 3 ]x1 + [2] >= [-& 3 ]x1 + [2] = a(a(b(b(b(x1)))))
       problem:
        DPs:
         a#(a(x1)) -> 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:
         a#(a(x1)) -> 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:
          a#(a(x1)) -> 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))) -> 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)))))
         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:
                     [0  -&]     [-&]
           [b](x0) = [2  -&]x0 + [0 ],
           
           [b#](x0) = [2 0]x0 + [0],
           
                     [0 0]     [0]
           [a](x0) = [2 0]x0 + [0],
           
           [a#](x0) = [0 0]x0 + [0]
          orientation:
           a#(a(x1)) = [2 0]x1 + [0] >= [2 0]x1 + [0] = b#(x1)
           
           b#(a(b(x1))) = [4  -&]x1 + [2] >= [2  -&]x1 + [0] = a#(a(b(b(b(x1)))))
           
                   [0 0]     [0]           
           a(x1) = [2 0]x1 + [0] >= x1 = x1
           
                      [2 0]     [0]    [0  -&]     [-&]        
           a(a(x1)) = [2 2]x1 + [2] >= [2  -&]x1 + [0 ] = b(x1)
           
                         [2  -&]     [0]    [2  -&]     [0]                    
           b(a(b(x1))) = [4  -&]x1 + [2] >= [4  -&]x1 + [2] = a(a(b(b(b(x1)))))
          problem:
           DPs:
            a#(a(x1)) -> b#(x1)
           TRS:
            a(x1) -> x1
            a(a(x1)) -> b(x1)
            b(a(b(x1))) -> a(a(b(b(b(x1)))))
          Restore Modifier:
           DPs:
            a#(a(x1)) -> b#(x1)
           TRS:
            a(x1) -> x1
            a(a(x1)) -> b(x1)
            b(a(b(x1))) -> a(a(b(b(b(x1)))))
           EDG Processor:
            DPs:
             a#(a(x1)) -> b#(x1)
            TRS:
             a(x1) -> x1
             a(a(x1)) -> b(x1)
             b(a(b(x1))) -> a(a(b(b(b(x1)))))
            graph:
             
            SCC Processor:
             #sccs: 0
             #rules: 0
             #arcs: 0/1