YES

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

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