YES

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

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