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)))
  Arctic Interpretation Processor:
   dimension: 1
   usable rules:
    a(a(x1)) -> b(b(b(x1)))
    b(b(b(b(b(x1))))) -> a(a(a(x1)))
   interpretation:
    [b#](x0) = x0,
    
    [a#](x0) = 3x0 + 1,
    
    [b](x0) = 5x0 + 1,
    
    [a](x0) = 8x0 + 4
   orientation:
    a#(a(x1)) = 11x1 + 7 >= x1 = b#(x1)
    
    a#(a(x1)) = 11x1 + 7 >= 5x1 + 1 = b#(b(x1))
    
    a#(a(x1)) = 11x1 + 7 >= 10x1 + 6 = b#(b(b(x1)))
    
    b#(b(b(b(b(x1))))) = 20x1 + 16 >= 3x1 + 1 = a#(x1)
    
    b#(b(b(b(b(x1))))) = 20x1 + 16 >= 11x1 + 7 = a#(a(x1))
    
    b#(b(b(b(b(x1))))) = 20x1 + 16 >= 19x1 + 15 = a#(a(a(x1)))
    
    a(a(x1)) = 16x1 + 12 >= 15x1 + 11 = b(b(b(x1)))
    
    b(b(b(b(b(x1))))) = 25x1 + 21 >= 24x1 + 20 = 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