YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(a(x1))) -> b#(x1)
   b#(b(x1)) -> a#(b(x1))
   b#(b(x1)) -> b#(a(b(x1)))
   b#(b(x1)) -> a#(x1)
   b#(b(x1)) -> a#(a(x1))
  TRS:
   a(a(a(x1))) -> b(x1)
   b(b(x1)) -> b(a(b(x1)))
   b(b(x1)) -> a(a(x1))
  TDG Processor:
   DPs:
    a#(a(a(x1))) -> b#(x1)
    b#(b(x1)) -> a#(b(x1))
    b#(b(x1)) -> b#(a(b(x1)))
    b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> a#(a(x1))
   TRS:
    a(a(a(x1))) -> b(x1)
    b(b(x1)) -> b(a(b(x1)))
    b(b(x1)) -> a(a(x1))
   graph:
    b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(a(x1))
    b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(x1)
    b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1)))
    b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(b(x1))
    b#(b(x1)) -> a#(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(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(a(x1))) -> b#(x1) -> b#(b(x1)) -> b#(a(b(x1)))
    a#(a(a(x1))) -> b#(x1) -> b#(b(x1)) -> a#(b(x1))
   Arctic Interpretation Processor:
    dimension: 2
    interpretation:
     [b#](x0) = [1 1]x0 + [0],
     
     [a#](x0) = [0 2]x0 + [0],
     
               [1 1]     [1]
     [b](x0) = [0 0]x0 + [0],
     
               [0  1 ]     [0]
     [a](x0) = [0  -&]x0 + [0]
    orientation:
     a#(a(a(x1))) = [2 3]x1 + [2] >= [1 1]x1 + [0] = b#(x1)
     
     b#(b(x1)) = [2 2]x1 + [2] >= [2 2]x1 + [2] = a#(b(x1))
     
     b#(b(x1)) = [2 2]x1 + [2] >= [2 2]x1 + [2] = b#(a(b(x1)))
     
     b#(b(x1)) = [2 2]x1 + [2] >= [0 2]x1 + [0] = a#(x1)
     
     b#(b(x1)) = [2 2]x1 + [2] >= [2 1]x1 + [2] = a#(a(x1))
     
                   [1 2]     [1]    [1 1]     [1]        
     a(a(a(x1))) = [1 1]x1 + [1] >= [0 0]x1 + [0] = b(x1)
     
                [2 2]     [2]    [2 2]     [2]              
     b(b(x1)) = [1 1]x1 + [1] >= [1 1]x1 + [1] = b(a(b(x1)))
     
                [2 2]     [2]    [1 1]     [1]           
     b(b(x1)) = [1 1]x1 + [1] >= [0 1]x1 + [0] = a(a(x1))
    problem:
     DPs:
      b#(b(x1)) -> a#(b(x1))
      b#(b(x1)) -> b#(a(b(x1)))
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> a#(a(x1))
     TRS:
      a(a(a(x1))) -> b(x1)
      b(b(x1)) -> b(a(b(x1)))
      b(b(x1)) -> a(a(x1))
    EDG Processor:
     DPs:
      b#(b(x1)) -> a#(b(x1))
      b#(b(x1)) -> b#(a(b(x1)))
      b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> a#(a(x1))
     TRS:
      a(a(a(x1))) -> b(x1)
      b(b(x1)) -> b(a(b(x1)))
      b(b(x1)) -> a(a(x1))
     graph:
      b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(b(x1))
      b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> b#(a(b(x1)))
      b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(x1)
      b#(b(x1)) -> b#(a(b(x1))) -> b#(b(x1)) -> a#(a(x1))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 4/16
      DPs:
       b#(b(x1)) -> b#(a(b(x1)))
      TRS:
       a(a(a(x1))) -> b(x1)
       b(b(x1)) -> b(a(b(x1)))
       b(b(x1)) -> a(a(x1))
      Arctic Interpretation Processor:
       dimension: 2
       interpretation:
        [b#](x0) = [-& 0 ]x0 + [0],
        
                  [0 2]     [0]
        [b](x0) = [1 3]x0 + [2],
        
                  [1  2 ]     [3]
        [a](x0) = [0  -&]x0 + [0]
       orientation:
        b#(b(x1)) = [1 3]x1 + [2] >= [0 2]x1 + [0] = b#(a(b(x1)))
        
                      [3 4]     [5]    [0 2]     [0]        
        a(a(a(x1))) = [2 3]x1 + [4] >= [1 3]x1 + [2] = b(x1)
        
                   [3 5]     [4]    [3 5]     [4]              
        b(b(x1)) = [4 6]x1 + [5] >= [4 6]x1 + [5] = b(a(b(x1)))
        
                   [3 5]     [4]    [2 3]     [4]           
        b(b(x1)) = [4 6]x1 + [5] >= [1 2]x1 + [3] = a(a(x1))
       problem:
        DPs:
         
        TRS:
         a(a(a(x1))) -> b(x1)
         b(b(x1)) -> b(a(b(x1)))
         b(b(x1)) -> a(a(x1))
       Qed