YES

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> b#(x1)
   a#(x1) -> b#(b(x1))
   b#(b(b(x1))) -> a#(x1)
  TRS:
   a(x1) -> b(b(x1))
   b(b(b(x1))) -> a(x1)
  TDG Processor:
   DPs:
    a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1))
    b#(b(b(x1))) -> a#(x1)
   TRS:
    a(x1) -> b(b(x1))
    b(b(b(x1))) -> a(x1)
   graph:
    b#(b(b(x1))) -> a#(x1) -> a#(x1) -> b#(b(x1))
    b#(b(b(x1))) -> a#(x1) -> a#(x1) -> b#(x1)
    a#(x1) -> b#(b(x1)) -> b#(b(b(x1))) -> a#(x1)
    a#(x1) -> b#(x1) -> b#(b(b(x1))) -> a#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = 3x0 + -16,
     
     [a#](x0) = 8x0 + 0,
     
     [b](x0) = 4x0 + -7,
     
     [a](x0) = 8x0 + 0
    orientation:
     a#(x1) = 8x1 + 0 >= 3x1 + -16 = b#(x1)
     
     a#(x1) = 8x1 + 0 >= 7x1 + -4 = b#(b(x1))
     
     b#(b(b(x1))) = 11x1 + 0 >= 8x1 + 0 = a#(x1)
     
     a(x1) = 8x1 + 0 >= 8x1 + -3 = b(b(x1))
     
     b(b(b(x1))) = 12x1 + 1 >= 8x1 + 0 = a(x1)
    problem:
     DPs:
      b#(b(b(x1))) -> a#(x1)
     TRS:
      a(x1) -> b(b(x1))
      b(b(b(x1))) -> a(x1)
    EDG Processor:
     DPs:
      b#(b(b(x1))) -> a#(x1)
     TRS:
      a(x1) -> b(b(x1))
      b(b(b(x1))) -> a(x1)
     graph:
      
     Qed