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
     usable rules:
      a(x1) -> x1
      a(a(x1)) -> b(a(b(a(x1))))
      b(b(b(b(x1)))) -> a(x1)
     interpretation:
                [1  0 ]  
      [b](x0) = [0  -&]x0,
      
      [b#](x0) = [0 0]x0,
      
                [0 1]  
      [a](x0) = [1 2]x0,
      
      [a#](x0) = [1 1]x0
     orientation:
      a#(a(x1)) = [2 3]x1 >= [1 2]x1 = b#(a(x1))
      
      a#(a(x1)) = [2 3]x1 >= [2 3]x1 = a#(b(a(x1)))
      
      a#(a(x1)) = [2 3]x1 >= [2 3]x1 = b#(a(b(a(x1))))
      
      b#(b(b(b(x1)))) = [3 2]x1 >= [1 1]x1 = a#(x1)
      
              [0 1]             
      a(x1) = [1 2]x1 >= x1 = x1
      
                 [2 3]      [2 3]                   
      a(a(x1)) = [3 4]x1 >= [1 2]x1 = b(a(b(a(x1))))
      
                       [4 3]      [0 1]          
      b(b(b(b(x1)))) = [3 2]x1 >= [1 2]x1 = 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)
     Restore Modifier:
      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)) -> a#(b(a(x1)))
        a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> b#(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
         usable rules:
          a(x1) -> x1
          a(a(x1)) -> b(a(b(a(x1))))
          b(b(b(b(x1)))) -> a(x1)
         interpretation:
                    [-& 0 ]     [0]
          [b](x0) = [0  1 ]x0 + [1],
          
                    [2 1]     [3]
          [a](x0) = [1 0]x0 + [0],
          
          [a#](x0) = [0  -&]x0 + [0]
         orientation:
          a#(a(x1)) = [2 1]x1 + [3] >= [1 0]x1 + [0] = a#(b(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]     [3]    [2 1]     [3]        
          b(b(b(b(x1)))) = [3 4]x1 + [4] >= [1 0]x1 + [0] = a(x1)
         problem:
          DPs:
           
          TRS:
           a(x1) -> x1
           a(a(x1)) -> b(a(b(a(x1))))
           b(b(b(b(x1)))) -> a(x1)
         Qed