YES

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

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