YES

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

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