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)))
  EDG 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))) -> 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))) -> a#(b(b(x)))
    b#(b(a(x))) -> b#(x) -> b#(b(a(x))) -> 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))) -> a#(b(b(x)))
    b#(b(a(x))) -> a#(b(b(x))) -> a#(a(x)) -> b#(x)
    b#(b(a(x))) -> a#(b(b(x))) -> a#(a(x)) -> b#(b(x))
    a#(a(x)) -> b#(b(x)) -> b#(b(a(x))) -> 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))) -> a#(b(b(x)))
    a#(a(x)) -> b#(x) -> b#(b(a(x))) -> b#(x)
    a#(a(x)) -> b#(x) -> b#(b(a(x))) -> b#(b(x))
    a#(a(x)) -> b#(x) -> b#(b(a(x))) -> a#(b(b(x)))
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = x0 + 1,
     
     [a#](x0) = x0 + 1,
     
     [b](x0) = x0,
     
     [a](x0) = x0 + 1
    orientation:
     a#(a(x)) = x + 2 >= x + 1 = b#(x)
     
     a#(a(x)) = x + 2 >= x + 1 = b#(b(x))
     
     b#(b(a(x))) = x + 2 >= x + 1 = b#(x)
     
     b#(b(a(x))) = x + 2 >= x + 1 = b#(b(x))
     
     b#(b(a(x))) = x + 2 >= x + 1 = a#(b(b(x)))
     
     a(a(x)) = x + 2 >= x = b(b(x))
     
     b(b(a(x))) = x + 1 >= x + 1 = a(b(b(x)))
    problem:
     DPs:
      
     TRS:
      a(a(x)) -> b(b(x))
      b(b(a(x))) -> a(b(b(x)))
    Qed