YES

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

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