YES

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

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