YES

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

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