YES

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

Proof:
 DP Processor:
  DPs:
   b#(b(b(x1))) -> a#(b(x1))
   a#(a(x1)) -> b#(a(x1))
   a#(a(x1)) -> a#(b(a(x1)))
   a#(a(a(x1))) -> b#(x1)
   a#(a(a(x1))) -> b#(b(x1))
   a#(a(a(x1))) -> a#(b(b(x1)))
  TRS:
   b(b(b(x1))) -> a(b(x1))
   a(a(x1)) -> a(b(a(x1)))
   a(a(a(x1))) -> a(b(b(x1)))
  TDG Processor:
   DPs:
    b#(b(b(x1))) -> a#(b(x1))
    a#(a(x1)) -> b#(a(x1))
    a#(a(x1)) -> a#(b(a(x1)))
    a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> b#(b(x1))
    a#(a(a(x1))) -> a#(b(b(x1)))
   TRS:
    b(b(b(x1))) -> a(b(x1))
    a(a(x1)) -> a(b(a(x1)))
    a(a(a(x1))) -> a(b(b(x1)))
   graph:
    a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1)))
    a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1))
    a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(x1)))
    a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(x1))
    a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(x1))
    a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(x1))
    a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> a#(b(b(x1)))
    a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(b(x1))
    a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(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))
    a#(a(x1)) -> b#(a(x1)) -> b#(b(b(x1))) -> a#(b(x1))
    b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(b(x1)))
    b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(b(x1))
    b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1)
    b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1)))
    b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(x1))
   EDG Processor:
    DPs:
     b#(b(b(x1))) -> a#(b(x1))
     a#(a(x1)) -> b#(a(x1))
     a#(a(x1)) -> a#(b(a(x1)))
     a#(a(a(x1))) -> b#(x1)
     a#(a(a(x1))) -> b#(b(x1))
     a#(a(a(x1))) -> a#(b(b(x1)))
    TRS:
     b(b(b(x1))) -> a(b(x1))
     a(a(x1)) -> a(b(a(x1)))
     a(a(a(x1))) -> a(b(b(x1)))
    graph:
     a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(x1))
     a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(x1)))
     a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1)
     a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1))
     a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1)))
     a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(x1))
     a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(x1))
     a#(a(x1)) -> a#(b(a(x1))) -> a#(a(x1)) -> 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(a(x1))) -> b#(x1)
     a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(b(x1))
     a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> a#(b(b(x1)))
     b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(x1))
     b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1)))
     b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1)
     b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(b(x1))
     b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(b(x1)))
    CDG Processor:
     DPs:
      b#(b(b(x1))) -> a#(b(x1))
      a#(a(x1)) -> b#(a(x1))
      a#(a(x1)) -> a#(b(a(x1)))
      a#(a(a(x1))) -> b#(x1)
      a#(a(a(x1))) -> b#(b(x1))
      a#(a(a(x1))) -> a#(b(b(x1)))
     TRS:
      b(b(b(x1))) -> a(b(x1))
      a(a(x1)) -> a(b(a(x1)))
      a(a(a(x1))) -> a(b(b(x1)))
     graph:
      a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1)))
      a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(b(x1))
      a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1))) -> b#(x1)
      a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> a#(b(a(x1)))
      a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1)) -> b#(a(x1))
      a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(b(x1))
      a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(x1))
      b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> a#(b(b(x1)))
      b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(b(x1))
      b#(b(b(x1))) -> a#(b(x1)) -> a#(a(a(x1))) -> b#(x1)
      b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(a(x1)))
      b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(x1))
     SCC Processor:
      #sccs: 1
      #rules: 4
      #arcs: 12/36
      DPs:
       a#(a(a(x1))) -> a#(b(b(x1)))
       a#(a(a(x1))) -> b#(x1)
       b#(b(b(x1))) -> a#(b(x1))
       a#(a(a(x1))) -> b#(b(x1))
      TRS:
       b(b(b(x1))) -> a(b(x1))
       a(a(x1)) -> a(b(a(x1)))
       a(a(a(x1))) -> a(b(b(x1)))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [a#](x0) = [3 0]x0 + [0],
        
        [b#](x0) = [0 2]x0,
        
                  [1  0 ]     [0]
        [a](x0) = [0  -&]x0 + [0],
        
                  [0 0]     [-&]
        [b](x0) = [1 0]x0 + [0 ]
       orientation:
        a#(a(a(x1))) = [5 4]x1 + [4] >= [4 3]x1 + [3] = a#(b(b(x1)))
        
        a#(a(a(x1))) = [5 4]x1 + [4] >= [0 2]x1 = b#(x1)
        
        b#(b(b(x1))) = [3 3]x1 + [2] >= [3 3]x1 + [0] = a#(b(x1))
        
        a#(a(a(x1))) = [5 4]x1 + [4] >= [3 2]x1 + [2] = b#(b(x1))
        
                      [1 1]     [0]    [1 1]     [0]           
        b(b(b(x1))) = [2 1]x1 + [1] >= [0 0]x1 + [0] = a(b(x1))
        
                   [2 1]     [1]    [2 1]     [1]              
        a(a(x1)) = [1 0]x1 + [0] >= [1 0]x1 + [0] = a(b(a(x1)))
        
                      [3 2]     [2]    [2 1]     [1]              
        a(a(a(x1))) = [2 1]x1 + [1] >= [1 0]x1 + [0] = a(b(b(x1)))
       problem:
        DPs:
         b#(b(b(x1))) -> a#(b(x1))
        TRS:
         b(b(b(x1))) -> a(b(x1))
         a(a(x1)) -> a(b(a(x1)))
         a(a(a(x1))) -> a(b(b(x1)))
       EDG Processor:
        DPs:
         b#(b(b(x1))) -> a#(b(x1))
        TRS:
         b(b(b(x1))) -> a(b(x1))
         a(a(x1)) -> a(b(a(x1)))
         a(a(a(x1))) -> a(b(b(x1)))
        graph:
         
        Qed