YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> b#(b(x1))
   a#(a(x1)) -> b#(b(b(x1)))
   b#(b(b(b(b(x1))))) -> a#(x1)
   b#(b(b(b(b(x1))))) -> a#(a(x1))
   b#(b(b(b(b(x1))))) -> a#(a(a(x1)))
  TRS:
   a(a(x1)) -> b(b(b(x1)))
   b(b(b(b(b(x1))))) -> a(a(a(x1)))
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(x1))
    a#(a(x1)) -> b#(b(b(x1)))
    b#(b(b(b(b(x1))))) -> a#(x1)
    b#(b(b(b(b(x1))))) -> a#(a(x1))
    b#(b(b(b(b(x1))))) -> a#(a(a(x1)))
   TRS:
    a(a(x1)) -> b(b(b(x1)))
    b(b(b(b(b(x1))))) -> a(a(a(x1)))
   graph:
    b#(b(b(b(b(x1))))) -> a#(a(a(x1))) ->
    a#(a(x1)) -> b#(b(b(x1)))
    b#(b(b(b(b(x1))))) -> a#(a(a(x1))) -> a#(a(x1)) -> b#(b(x1))
    b#(b(b(b(b(x1))))) -> a#(a(a(x1))) -> a#(a(x1)) -> b#(x1)
    b#(b(b(b(b(x1))))) -> a#(a(x1)) -> a#(a(x1)) -> b#(b(b(x1)))
    b#(b(b(b(b(x1))))) -> a#(a(x1)) -> a#(a(x1)) -> b#(b(x1))
    b#(b(b(b(b(x1))))) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
    b#(b(b(b(b(x1))))) -> a#(x1) -> a#(a(x1)) -> b#(b(b(x1)))
    b#(b(b(b(b(x1))))) -> a#(x1) -> a#(a(x1)) -> b#(b(x1))
    b#(b(b(b(b(x1))))) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(b(b(b(x1))))) -> a#(a(a(x1)))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(b(b(b(x1))))) -> a#(a(x1))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(b(b(b(x1))))) -> a#(x1)
    a#(a(x1)) -> b#(b(x1)) -> b#(b(b(b(b(x1))))) -> a#(a(a(x1)))
    a#(a(x1)) -> b#(b(x1)) -> b#(b(b(b(b(x1))))) -> a#(a(x1))
    a#(a(x1)) -> b#(b(x1)) -> b#(b(b(b(b(x1))))) -> a#(x1)
    a#(a(x1)) -> b#(x1) -> b#(b(b(b(b(x1))))) -> a#(a(a(x1)))
    a#(a(x1)) -> b#(x1) -> b#(b(b(b(b(x1))))) -> a#(a(x1))
    a#(a(x1)) -> b#(x1) -> b#(b(b(b(b(x1))))) -> a#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = 3x0,
     
     [a#](x0) = 6x0,
     
     [b](x0) = 5x0,
     
     [a](x0) = 8x0
    orientation:
     a#(a(x1)) = 14x1 >= 3x1 = b#(x1)
     
     a#(a(x1)) = 14x1 >= 8x1 = b#(b(x1))
     
     a#(a(x1)) = 14x1 >= 13x1 = b#(b(b(x1)))
     
     b#(b(b(b(b(x1))))) = 23x1 >= 6x1 = a#(x1)
     
     b#(b(b(b(b(x1))))) = 23x1 >= 14x1 = a#(a(x1))
     
     b#(b(b(b(b(x1))))) = 23x1 >= 22x1 = a#(a(a(x1)))
     
     a(a(x1)) = 16x1 >= 15x1 = b(b(b(x1)))
     
     b(b(b(b(b(x1))))) = 25x1 >= 24x1 = a(a(a(x1)))
    problem:
     DPs:
      
     TRS:
      a(a(x1)) -> b(b(b(x1)))
      b(b(b(b(b(x1))))) -> a(a(a(x1)))
    Qed