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