YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(a(x1))) -> b#(x1)
   a#(a(a(x1))) -> b#(b(x1))
   a#(a(a(x1))) -> b#(b(b(x1)))
   a#(a(a(x1))) -> a#(b(b(b(x1))))
   b#(b(a(b(x1)))) -> a#(x1)
   b#(b(a(b(x1)))) -> a#(a(x1))
   b#(b(a(b(x1)))) -> a#(a(a(x1)))
  TRS:
   a(a(a(x1))) -> a(b(b(b(x1))))
   b(b(a(b(x1)))) -> a(a(a(x1)))
  TDG Processor:
   DPs:
    a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> b#(b(x1))
    a#(a(a(x1))) -> b#(b(b(x1)))
    a#(a(a(x1))) -> a#(b(b(b(x1))))
    b#(b(a(b(x1)))) -> a#(x1)
    b#(b(a(b(x1)))) -> a#(a(x1))
    b#(b(a(b(x1)))) -> a#(a(a(x1)))
   TRS:
    a(a(a(x1))) -> a(b(b(b(x1))))
    b(b(a(b(x1)))) -> a(a(a(x1)))
   graph:
    b#(b(a(b(x1)))) -> a#(a(a(x1))) ->
    a#(a(a(x1))) -> a#(b(b(b(x1))))
    b#(b(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(x1))) -> b#(b(b(x1)))
    b#(b(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(x1))) -> b#(b(x1))
    b#(b(a(b(x1)))) -> a#(a(a(x1))) -> a#(a(a(x1))) -> b#(x1)
    b#(b(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> a#(b(b(b(x1))))
    b#(b(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(b(x1)))
    b#(b(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1))
    b#(b(a(b(x1)))) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1)
    b#(b(a(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> a#(b(b(b(x1))))
    b#(b(a(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(b(x1)))
    b#(b(a(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1))
    b#(b(a(b(x1)))) -> a#(x1) -> a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> a#(a(a(x1)))
    a#(a(a(x1))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> a#(a(x1))
    a#(a(a(x1))) -> b#(b(b(x1))) -> b#(b(a(b(x1)))) -> a#(x1)
    a#(a(a(x1))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> a#(a(a(x1)))
    a#(a(a(x1))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> a#(a(x1))
    a#(a(a(x1))) -> b#(b(x1)) -> b#(b(a(b(x1)))) -> a#(x1)
    a#(a(a(x1))) -> b#(x1) -> b#(b(a(b(x1)))) -> a#(a(a(x1)))
    a#(a(a(x1))) -> b#(x1) -> b#(b(a(b(x1)))) -> a#(a(x1))
    a#(a(a(x1))) -> b#(x1) -> b#(b(a(b(x1)))) -> a#(x1)
    a#(a(a(x1))) -> a#(b(b(b(x1)))) ->
    a#(a(a(x1))) -> a#(b(b(b(x1))))
    a#(a(a(x1))) -> a#(b(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(b(x1)))
    a#(a(a(x1))) -> a#(b(b(b(x1)))) -> a#(a(a(x1))) -> b#(b(x1))
    a#(a(a(x1))) -> a#(b(b(b(x1)))) -> a#(a(a(x1))) -> b#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = x0,
     
     [a#](x0) = 1x0,
     
     [b](x0) = 2x0,
     
     [a](x0) = 3x0
    orientation:
     a#(a(a(x1))) = 7x1 >= x1 = b#(x1)
     
     a#(a(a(x1))) = 7x1 >= 2x1 = b#(b(x1))
     
     a#(a(a(x1))) = 7x1 >= 4x1 = b#(b(b(x1)))
     
     a#(a(a(x1))) = 7x1 >= 7x1 = a#(b(b(b(x1))))
     
     b#(b(a(b(x1)))) = 7x1 >= 1x1 = a#(x1)
     
     b#(b(a(b(x1)))) = 7x1 >= 4x1 = a#(a(x1))
     
     b#(b(a(b(x1)))) = 7x1 >= 7x1 = a#(a(a(x1)))
     
     a(a(a(x1))) = 9x1 >= 9x1 = a(b(b(b(x1))))
     
     b(b(a(b(x1)))) = 9x1 >= 9x1 = a(a(a(x1)))
    problem:
     DPs:
      a#(a(a(x1))) -> a#(b(b(b(x1))))
      b#(b(a(b(x1)))) -> a#(a(a(x1)))
     TRS:
      a(a(a(x1))) -> a(b(b(b(x1))))
      b(b(a(b(x1)))) -> a(a(a(x1)))
    EDG Processor:
     DPs:
      a#(a(a(x1))) -> a#(b(b(b(x1))))
      b#(b(a(b(x1)))) -> a#(a(a(x1)))
     TRS:
      a(a(a(x1))) -> a(b(b(b(x1))))
      b(b(a(b(x1)))) -> a(a(a(x1)))
     graph:
      b#(b(a(b(x1)))) -> a#(a(a(x1))) ->
      a#(a(a(x1))) -> a#(b(b(b(x1))))
      a#(a(a(x1))) -> a#(b(b(b(x1)))) -> a#(a(a(x1))) -> a#(b(b(b(x1))))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 2/4
      DPs:
       a#(a(a(x1))) -> a#(b(b(b(x1))))
      TRS:
       a(a(a(x1))) -> a(b(b(b(x1))))
       b(b(a(b(x1)))) -> a(a(a(x1)))
      Arctic Interpretation Processor:
       dimension: 4
       interpretation:
        [a#](x0) = [-& -& 0  -&]x0 + [0],
        
                  [-& 0  0  0 ]     [0]
                  [0  -& -& 0 ]     [0]
        [b](x0) = [0  -& -& -&]x0 + [0]
                  [0  -& -& 0 ]     [0],
        
                  [0  -& -& 0 ]     [0]
                  [1  0  0  1 ]     [1]
        [a](x0) = [0  0  0  0 ]x0 + [0]
                  [0  -& 0  0 ]     [0]
       orientation:
        a#(a(a(x1))) = [1 0 0 1]x1 + [1] >= [0  -& -& 0 ]x1 + [0] = a#(b(b(b(x1))))
        
                      [0 0 0 0]     [0]    [0 0 0 0]     [0]                 
                      [1 1 1 1]     [1]    [1 1 1 1]     [1]                 
        a(a(a(x1))) = [1 0 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = a(b(b(b(x1))))
                      [1 0 0 1]     [1]    [0 0 0 0]     [0]                 
        
                         [0 0 0 0]     [0]    [0 0 0 0]     [0]              
                         [1 1 1 1]     [1]    [1 1 1 1]     [1]              
        b(b(a(b(x1)))) = [1 1 1 1]x1 + [1] >= [1 0 1 1]x1 + [1] = a(a(a(x1)))
                         [1 1 1 1]     [1]    [1 0 0 1]     [1]              
       problem:
        DPs:
         
        TRS:
         a(a(a(x1))) -> a(b(b(b(x1))))
         b(b(a(b(x1)))) -> a(a(a(x1)))
       Qed