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
    interpretation:
     [b#](x0) = [2 0]x0 + [0],
     
     [a#](x0) = [-& 0 ]x0 + [0],
     
               [1 2]     [2]
     [b](x0) = [2 0]x0 + [0],
     
               [-& 0 ]     [0]
     [a](x0) = [0  2 ]x0 + [2]
    orientation:
     a#(a(a(x1))) = [2 4]x1 + [4] >= [2 0]x1 + [0] = b#(x1)
     
     b#(b(x1)) = [3 4]x1 + [4] >= [-& 0 ]x1 + [0] = a#(x1)
     
     b#(b(x1)) = [3 4]x1 + [4] >= [0 2]x1 + [2] = a#(a(x1))
     
     a#(a(x1)) = [0 2]x1 + [2] >= [0 2]x1 + [2] = b#(a(x1))
     
     a#(a(x1)) = [0 2]x1 + [2] >= [0 2]x1 + [2] = a#(b(a(x1)))
     
                   [2 4]     [4]    [1 2]     [2]        
     a(a(a(x1))) = [4 6]x1 + [6] >= [2 0]x1 + [0] = b(x1)
     
                [4 3]     [3]    [0 2]     [2]           
     b(b(x1)) = [3 4]x1 + [4] >= [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:
      a#(a(a(x1))) -> b#(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:
      a#(a(a(x1))) -> b#(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:
      a#(a(x1)) -> a#(b(a(x1))) -> a#(a(a(x1))) -> b#(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)))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 3/9
      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
       interpretation:
        [a#](x0) = [-2 2 ]x0 + [0],
        
                  [-4 3 ]     [3 ]
        [b](x0) = [1  -2]x0 + [-4],
        
                  [-& -1]     [0]
        [a](x0) = [-3 2 ]x0 + [2]
       orientation:
        a#(a(x1)) = [-1 4 ]x1 + [4] >= [-2 3 ]x1 + [3] = a#(b(a(x1)))
        
                      [-2 3 ]     [3]    [-4 3 ]     [3 ]        
        a(a(a(x1))) = [1  6 ]x1 + [6] >= [1  -2]x1 + [-4] = b(x1)
        
                   [4  1 ]     [3]    [-4 1 ]     [1]           
        b(b(x1)) = [-1 4 ]x1 + [4] >= [-1 4 ]x1 + [4] = a(a(x1))
        
                   [-4 1 ]     [1]    [-6 -1]     [0]              
        a(a(x1)) = [-1 4 ]x1 + [4] >= [-3 2 ]x1 + [3] = 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