YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x)) -> b#(x)
   a#(a(x)) -> b#(b(x))
   b#(b(a(x))) -> b#(x)
   b#(b(a(x))) -> b#(b(x))
   b#(b(a(x))) -> a#(b(b(x)))
  TRS:
   a(a(x)) -> b(b(x))
   b(b(a(x))) -> a(b(b(x)))
  TDG Processor:
   DPs:
    a#(a(x)) -> b#(x)
    a#(a(x)) -> b#(b(x))
    b#(b(a(x))) -> b#(x)
    b#(b(a(x))) -> b#(b(x))
    b#(b(a(x))) -> a#(b(b(x)))
   TRS:
    a(a(x)) -> b(b(x))
    b(b(a(x))) -> a(b(b(x)))
   graph:
    b#(b(a(x))) -> b#(b(x)) -> b#(b(a(x))) -> a#(b(b(x)))
    b#(b(a(x))) -> b#(b(x)) -> b#(b(a(x))) -> b#(b(x))
    b#(b(a(x))) -> b#(b(x)) -> b#(b(a(x))) -> b#(x)
    b#(b(a(x))) -> b#(x) -> b#(b(a(x))) -> a#(b(b(x)))
    b#(b(a(x))) -> b#(x) -> b#(b(a(x))) -> b#(b(x))
    b#(b(a(x))) -> b#(x) -> b#(b(a(x))) -> b#(x)
    b#(b(a(x))) -> a#(b(b(x))) -> a#(a(x)) -> b#(b(x))
    b#(b(a(x))) -> a#(b(b(x))) -> a#(a(x)) -> b#(x)
    a#(a(x)) -> b#(b(x)) -> b#(b(a(x))) -> a#(b(b(x)))
    a#(a(x)) -> b#(b(x)) -> b#(b(a(x))) -> b#(b(x))
    a#(a(x)) -> b#(b(x)) -> b#(b(a(x))) -> b#(x)
    a#(a(x)) -> b#(x) -> b#(b(a(x))) -> a#(b(b(x)))
    a#(a(x)) -> b#(x) -> b#(b(a(x))) -> b#(b(x))
    a#(a(x)) -> b#(x) -> b#(b(a(x))) -> b#(x)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = x0,
     
     [a#](x0) = x0,
     
     [b](x0) = 8x0,
     
     [a](x0) = 8x0
    orientation:
     a#(a(x)) = 8x >= x = b#(x)
     
     a#(a(x)) = 8x >= 8x = b#(b(x))
     
     b#(b(a(x))) = 16x >= x = b#(x)
     
     b#(b(a(x))) = 16x >= 8x = b#(b(x))
     
     b#(b(a(x))) = 16x >= 16x = a#(b(b(x)))
     
     a(a(x)) = 16x >= 16x = b(b(x))
     
     b(b(a(x))) = 24x >= 24x = a(b(b(x)))
    problem:
     DPs:
      a#(a(x)) -> b#(b(x))
      b#(b(a(x))) -> a#(b(b(x)))
     TRS:
      a(a(x)) -> b(b(x))
      b(b(a(x))) -> a(b(b(x)))
    EDG Processor:
     DPs:
      a#(a(x)) -> b#(b(x))
      b#(b(a(x))) -> a#(b(b(x)))
     TRS:
      a(a(x)) -> b(b(x))
      b(b(a(x))) -> a(b(b(x)))
     graph:
      b#(b(a(x))) -> a#(b(b(x))) -> a#(a(x)) -> b#(b(x))
      a#(a(x)) -> b#(b(x)) -> b#(b(a(x))) -> a#(b(b(x)))
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [b#](x0) = 4x0,
       
       [a#](x0) = 2x0 + 0,
       
       [b](x0) = x0,
       
       [a](x0) = 3x0 + -4
      orientation:
       a#(a(x)) = 5x + 0 >= 4x = b#(b(x))
       
       b#(b(a(x))) = 7x + 0 >= 2x + 0 = a#(b(b(x)))
       
       a(a(x)) = 6x + -1 >= x = b(b(x))
       
       b(b(a(x))) = 3x + -4 >= 3x + -4 = a(b(b(x)))
      problem:
       DPs:
        b#(b(a(x))) -> a#(b(b(x)))
       TRS:
        a(a(x)) -> b(b(x))
        b(b(a(x))) -> a(b(b(x)))
      EDG Processor:
       DPs:
        b#(b(a(x))) -> a#(b(b(x)))
       TRS:
        a(a(x)) -> b(b(x))
        b(b(a(x))) -> a(b(b(x)))
       graph:
        
       Qed