YES

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

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