YES

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

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