YES

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

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