YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(a(x1))) -> b#(a(x1))
   a#(a(a(x1))) -> b#(b(a(x1)))
   a#(b(a(x1))) -> b#(b(a(x1)))
   b#(a(b(x1))) -> a#(a(b(x1)))
  TRS:
   a(a(a(x1))) -> b(b(a(x1)))
   a(b(a(x1))) -> b(b(a(x1)))
   b(a(b(x1))) -> a(a(b(x1)))
  TDG Processor:
   DPs:
    a#(a(a(x1))) -> b#(a(x1))
    a#(a(a(x1))) -> b#(b(a(x1)))
    a#(b(a(x1))) -> b#(b(a(x1)))
    b#(a(b(x1))) -> a#(a(b(x1)))
   TRS:
    a(a(a(x1))) -> b(b(a(x1)))
    a(b(a(x1))) -> b(b(a(x1)))
    b(a(b(x1))) -> a(a(b(x1)))
   graph:
    b#(a(b(x1))) -> a#(a(b(x1))) -> a#(b(a(x1))) -> b#(b(a(x1)))
    b#(a(b(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(b(a(x1)))
    b#(a(b(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(a(x1))
    a#(b(a(x1))) -> b#(b(a(x1))) -> b#(a(b(x1))) -> a#(a(b(x1)))
    a#(a(a(x1))) -> b#(b(a(x1))) -> b#(a(b(x1))) -> a#(a(b(x1)))
    a#(a(a(x1))) -> b#(a(x1)) -> b#(a(b(x1))) -> a#(a(b(x1)))
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = 1x0,
     
     [a#](x0) = 1x0,
     
     [b](x0) = 1x0,
     
     [a](x0) = 1x0
    orientation:
     a#(a(a(x1))) = 3x1 >= 2x1 = b#(a(x1))
     
     a#(a(a(x1))) = 3x1 >= 3x1 = b#(b(a(x1)))
     
     a#(b(a(x1))) = 3x1 >= 3x1 = b#(b(a(x1)))
     
     b#(a(b(x1))) = 3x1 >= 3x1 = a#(a(b(x1)))
     
     a(a(a(x1))) = 3x1 >= 3x1 = b(b(a(x1)))
     
     a(b(a(x1))) = 3x1 >= 3x1 = b(b(a(x1)))
     
     b(a(b(x1))) = 3x1 >= 3x1 = a(a(b(x1)))
    problem:
     DPs:
      a#(a(a(x1))) -> b#(b(a(x1)))
      a#(b(a(x1))) -> b#(b(a(x1)))
      b#(a(b(x1))) -> a#(a(b(x1)))
     TRS:
      a(a(a(x1))) -> b(b(a(x1)))
      a(b(a(x1))) -> b(b(a(x1)))
      b(a(b(x1))) -> a(a(b(x1)))
    EDG Processor:
     DPs:
      a#(a(a(x1))) -> b#(b(a(x1)))
      a#(b(a(x1))) -> b#(b(a(x1)))
      b#(a(b(x1))) -> a#(a(b(x1)))
     TRS:
      a(a(a(x1))) -> b(b(a(x1)))
      a(b(a(x1))) -> b(b(a(x1)))
      b(a(b(x1))) -> a(a(b(x1)))
     graph:
      b#(a(b(x1))) -> a#(a(b(x1))) -> a#(a(a(x1))) -> b#(b(a(x1)))
      b#(a(b(x1))) -> a#(a(b(x1))) -> a#(b(a(x1))) -> b#(b(a(x1)))
      a#(b(a(x1))) -> b#(b(a(x1))) -> b#(a(b(x1))) -> a#(a(b(x1)))
      a#(a(a(x1))) -> b#(b(a(x1))) -> b#(a(b(x1))) -> a#(a(b(x1)))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [b#](x0) = [1  -&]x0,
       
       [a#](x0) = [1 2]x0,
       
                 [0  -&]  
       [b](x0) = [1  0 ]x0,
       
                 [0 3]  
       [a](x0) = [1 0]x0
      orientation:
       a#(a(a(x1))) = [5 6]x1 >= [1 4]x1 = b#(b(a(x1)))
       
       a#(b(a(x1))) = [3 6]x1 >= [1 4]x1 = b#(b(a(x1)))
       
       b#(a(b(x1))) = [5 4]x1 >= [5 4]x1 = a#(a(b(x1)))
       
                     [4 7]      [0 3]                
       a(a(a(x1))) = [5 4]x1 >= [1 4]x1 = b(b(a(x1)))
       
                     [4 7]      [0 3]                
       a(b(a(x1))) = [1 4]x1 >= [1 4]x1 = b(b(a(x1)))
       
                     [4 3]      [4 3]                
       b(a(b(x1))) = [5 4]x1 >= [5 4]x1 = a(a(b(x1)))
      problem:
       DPs:
        b#(a(b(x1))) -> a#(a(b(x1)))
       TRS:
        a(a(a(x1))) -> b(b(a(x1)))
        a(b(a(x1))) -> b(b(a(x1)))
        b(a(b(x1))) -> a(a(b(x1)))
      EDG Processor:
       DPs:
        b#(a(b(x1))) -> a#(a(b(x1)))
       TRS:
        a(a(a(x1))) -> b(b(a(x1)))
        a(b(a(x1))) -> b(b(a(x1)))
        b(a(b(x1))) -> a(a(b(x1)))
       graph:
        
       Qed