YES

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

Proof:
 DP Processor:
  DPs:
   b#(b(b(x1))) -> a#(b(x1))
   b#(b(b(x1))) -> a#(a(b(x1)))
   b#(a(a(x1))) -> b#(x1)
   b#(a(a(x1))) -> a#(b(x1))
   b#(a(a(x1))) -> a#(a(b(x1)))
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> a#(b(x1))
   a#(a(x1)) -> b#(a(b(x1)))
  TRS:
   b(b(b(x1))) -> a(a(b(x1)))
   b(a(a(x1))) -> a(a(b(x1)))
   a(a(x1)) -> b(a(b(x1)))
  TDG Processor:
   DPs:
    b#(b(b(x1))) -> a#(b(x1))
    b#(b(b(x1))) -> a#(a(b(x1)))
    b#(a(a(x1))) -> b#(x1)
    b#(a(a(x1))) -> a#(b(x1))
    b#(a(a(x1))) -> a#(a(b(x1)))
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> a#(b(x1))
    a#(a(x1)) -> b#(a(b(x1)))
   TRS:
    b(b(b(x1))) -> a(a(b(x1)))
    b(a(a(x1))) -> a(a(b(x1)))
    a(a(x1)) -> b(a(b(x1)))
   graph:
    a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(x1)))
    a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1))
    a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) -> a#(a(b(x1)))
    a#(a(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) -> a#(b(x1))
    a#(a(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) -> b#(x1)
    a#(a(x1)) -> b#(a(b(x1))) -> b#(b(b(x1))) -> a#(a(b(x1)))
    a#(a(x1)) -> b#(a(b(x1))) -> b#(b(b(x1))) -> a#(b(x1))
    a#(a(x1)) -> b#(x1) -> b#(a(a(x1))) -> a#(a(b(x1)))
    a#(a(x1)) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1))
    a#(a(x1)) -> b#(x1) -> b#(a(a(x1))) -> b#(x1)
    a#(a(x1)) -> b#(x1) -> b#(b(b(x1))) -> a#(a(b(x1)))
    a#(a(x1)) -> b#(x1) -> b#(b(b(x1))) -> a#(b(x1))
    b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(a(b(x1)))
    b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(x1))
    b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(x1)
    b#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(x1)))
    b#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1))
    b#(a(a(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
    b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(a(b(x1)))
    b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> a#(b(x1))
    b#(a(a(x1))) -> b#(x1) -> b#(a(a(x1))) -> b#(x1)
    b#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(a(b(x1)))
    b#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> a#(b(x1))
    b#(b(b(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(a(b(x1)))
    b#(b(b(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(x1))
    b#(b(b(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(x1)
    b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(x1)))
    b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1))
    b#(b(b(x1))) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
     [a#](x0) = [0 0]x0 + [0],
     
     [b#](x0) = [0  -&]x0 + [0],
     
               [0 1]     [0]
     [a](x0) = [1 0]x0 + [1],
     
               [1  0 ]     [0 ]
     [b](x0) = [0  -&]x0 + [-&]
    orientation:
     b#(b(b(x1))) = [2 1]x1 + [1] >= [1 0]x1 + [0] = a#(b(x1))
     
     b#(b(b(x1))) = [2 1]x1 + [1] >= [2 1]x1 + [1] = a#(a(b(x1)))
     
     b#(a(a(x1))) = [2 1]x1 + [2] >= [0  -&]x1 + [0] = b#(x1)
     
     b#(a(a(x1))) = [2 1]x1 + [2] >= [1 0]x1 + [0] = a#(b(x1))
     
     b#(a(a(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [1] = a#(a(b(x1)))
     
     a#(a(x1)) = [1 1]x1 + [1] >= [0  -&]x1 + [0] = b#(x1)
     
     a#(a(x1)) = [1 1]x1 + [1] >= [1 0]x1 + [0] = a#(b(x1))
     
     a#(a(x1)) = [1 1]x1 + [1] >= [1 0]x1 + [0] = b#(a(b(x1)))
     
                   [3 2]     [2]    [3 2]     [2]              
     b(b(b(x1))) = [2 1]x1 + [1] >= [2 1]x1 + [1] = a(a(b(x1)))
     
                   [3 2]     [3]    [3 2]     [2]              
     b(a(a(x1))) = [2 1]x1 + [2] >= [2 1]x1 + [1] = a(a(b(x1)))
     
                [2 1]     [2]    [2 1]     [1]              
     a(a(x1)) = [1 2]x1 + [1] >= [1 0]x1 + [0] = b(a(b(x1)))
    problem:
     DPs:
      b#(b(b(x1))) -> a#(a(b(x1)))
      b#(a(a(x1))) -> a#(a(b(x1)))
      a#(a(x1)) -> a#(b(x1))
      a#(a(x1)) -> b#(a(b(x1)))
     TRS:
      b(b(b(x1))) -> a(a(b(x1)))
      b(a(a(x1))) -> a(a(b(x1)))
      a(a(x1)) -> b(a(b(x1)))
    EDG Processor:
     DPs:
      b#(b(b(x1))) -> a#(a(b(x1)))
      b#(a(a(x1))) -> a#(a(b(x1)))
      a#(a(x1)) -> a#(b(x1))
      a#(a(x1)) -> b#(a(b(x1)))
     TRS:
      b(b(b(x1))) -> a(a(b(x1)))
      b(a(a(x1))) -> a(a(b(x1)))
      a(a(x1)) -> b(a(b(x1)))
     graph:
      a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(x1))
      a#(a(x1)) -> a#(b(x1)) -> a#(a(x1)) -> b#(a(b(x1)))
      a#(a(x1)) -> b#(a(b(x1))) -> b#(b(b(x1))) -> a#(a(b(x1)))
      a#(a(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) -> a#(a(b(x1)))
      b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(x1))
      b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(a(b(x1)))
      b#(b(b(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(x1))
      b#(b(b(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(a(b(x1)))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [a#](x0) = [0  -&]x0 + [0],
       
       [b#](x0) = [-& 0 ]x0 + [0],
       
                 [0 1]     [1 ]
       [a](x0) = [1 0]x0 + [-&],
       
                 [-& 0 ]     [0]
       [b](x0) = [0  1 ]x0 + [1]
      orientation:
       b#(b(b(x1))) = [1 2]x1 + [2] >= [1 2]x1 + [2] = a#(a(b(x1)))
       
       b#(a(a(x1))) = [1 2]x1 + [2] >= [1 2]x1 + [2] = a#(a(b(x1)))
       
       a#(a(x1)) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = a#(b(x1))
       
       a#(a(x1)) = [0 1]x1 + [1] >= [0 1]x1 + [1] = b#(a(b(x1)))
       
                     [1 2]     [2]    [1 2]     [2]              
       b(b(b(x1))) = [2 3]x1 + [3] >= [2 3]x1 + [3] = a(a(b(x1)))
       
                     [1 2]     [2]    [1 2]     [2]              
       b(a(a(x1))) = [2 3]x1 + [3] >= [2 3]x1 + [3] = a(a(b(x1)))
       
                  [2 1]     [1]    [0 1]     [1]              
       a(a(x1)) = [1 2]x1 + [2] >= [1 2]x1 + [2] = b(a(b(x1)))
      problem:
       DPs:
        b#(b(b(x1))) -> a#(a(b(x1)))
        b#(a(a(x1))) -> a#(a(b(x1)))
        a#(a(x1)) -> b#(a(b(x1)))
       TRS:
        b(b(b(x1))) -> a(a(b(x1)))
        b(a(a(x1))) -> a(a(b(x1)))
        a(a(x1)) -> b(a(b(x1)))
      EDG Processor:
       DPs:
        b#(b(b(x1))) -> a#(a(b(x1)))
        b#(a(a(x1))) -> a#(a(b(x1)))
        a#(a(x1)) -> b#(a(b(x1)))
       TRS:
        b(b(b(x1))) -> a(a(b(x1)))
        b(a(a(x1))) -> a(a(b(x1)))
        a(a(x1)) -> b(a(b(x1)))
       graph:
        a#(a(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) -> a#(a(b(x1)))
        a#(a(x1)) -> b#(a(b(x1))) -> b#(b(b(x1))) -> a#(a(b(x1)))
        b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(a(b(x1)))
        b#(b(b(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(a(b(x1)))
       Arctic Interpretation Processor:
        dimension: 2
        interpretation:
         [a#](x0) = [0 1]x0 + [3],
         
         [b#](x0) = [-& 3 ]x0 + [0],
         
                   [-& 3 ]     [2 ]
         [a](x0) = [-1 -2]x0 + [-&],
         
                   [-& 1 ]     [0]
         [b](x0) = [-3 2 ]x0 + [1]
        orientation:
         b#(b(b(x1))) = [2 7]x1 + [6] >= [0 5]x1 + [4] = a#(a(b(x1)))
         
         b#(a(a(x1))) = [0 5]x1 + [4] >= [0 5]x1 + [4] = a#(a(b(x1)))
         
         a#(a(x1)) = [0 3]x1 + [3] >= [-2 3 ]x1 + [2] = b#(a(b(x1)))
         
                       [0 5]     [4]    [-2 3 ]     [2]              
         b(b(b(x1))) = [1 6]x1 + [5] >= [-1 4 ]x1 + [3] = a(a(b(x1)))
         
                       [-2 3 ]     [2]    [-2 3 ]     [2]              
         b(a(a(x1))) = [-1 4 ]x1 + [3] >= [-1 4 ]x1 + [3] = a(a(b(x1)))
         
                    [2  1 ]     [2]    [-4 1 ]     [0]              
         a(a(x1)) = [-3 2 ]x1 + [1] >= [-3 2 ]x1 + [1] = b(a(b(x1)))
        problem:
         DPs:
          b#(a(a(x1))) -> a#(a(b(x1)))
          a#(a(x1)) -> b#(a(b(x1)))
         TRS:
          b(b(b(x1))) -> a(a(b(x1)))
          b(a(a(x1))) -> a(a(b(x1)))
          a(a(x1)) -> b(a(b(x1)))
        EDG Processor:
         DPs:
          b#(a(a(x1))) -> a#(a(b(x1)))
          a#(a(x1)) -> b#(a(b(x1)))
         TRS:
          b(b(b(x1))) -> a(a(b(x1)))
          b(a(a(x1))) -> a(a(b(x1)))
          a(a(x1)) -> b(a(b(x1)))
         graph:
          a#(a(x1)) -> b#(a(b(x1))) -> b#(a(a(x1))) -> a#(a(b(x1)))
          b#(a(a(x1))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(a(b(x1)))
         Arctic Interpretation Processor:
          dimension: 2
          interpretation:
           [a#](x0) = [0 2]x0 + [3],
           
           [b#](x0) = [2 0]x0 + [0],
           
                     [-3 3 ]     [0]
           [a](x0) = [0  -&]x0 + [0],
           
                     [2  0 ]     [2 ]
           [b](x0) = [-3 -&]x0 + [-3]
          orientation:
           b#(a(a(x1))) = [5 3]x1 + [5] >= [4 2]x1 + [4] = a#(a(b(x1)))
           
           a#(a(x1)) = [2 3]x1 + [3] >= [2 0]x1 + [2] = b#(a(b(x1)))
           
                         [6  4 ]     [6]    [5  3 ]     [5]              
           b(b(b(x1))) = [1  -1]x1 + [1] >= [0  -3]x1 + [0] = a(a(b(x1)))
           
                         [5  3 ]     [5]    [5  3 ]     [5]              
           b(a(a(x1))) = [0  -3]x1 + [0] >= [0  -3]x1 + [0] = a(a(b(x1)))
           
                      [3  0 ]     [3]    [2  0 ]     [2 ]              
           a(a(x1)) = [-3 3 ]x1 + [0] >= [-3 -6]x1 + [-3] = b(a(b(x1)))
          problem:
           DPs:
            a#(a(x1)) -> b#(a(b(x1)))
           TRS:
            b(b(b(x1))) -> a(a(b(x1)))
            b(a(a(x1))) -> a(a(b(x1)))
            a(a(x1)) -> b(a(b(x1)))
          EDG Processor:
           DPs:
            a#(a(x1)) -> b#(a(b(x1)))
           TRS:
            b(b(b(x1))) -> a(a(b(x1)))
            b(a(a(x1))) -> a(a(b(x1)))
            a(a(x1)) -> b(a(b(x1)))
           graph:
            
           Qed