YES

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

Proof:
 String Reversal Processor:
  a(x1) -> x1
  b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
  DP Processor:
   DPs:
    b#(b(a(a(x1)))) -> b#(x1)
    b#(b(a(a(x1)))) -> b#(b(x1))
    b#(b(a(a(x1)))) -> a#(b(b(x1)))
    b#(b(a(a(x1)))) -> a#(a(b(b(x1))))
    b#(b(a(a(x1)))) -> b#(a(a(b(b(x1)))))
    b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1))))))
   TRS:
    a(x1) -> x1
    b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
   TDG Processor:
    DPs:
     b#(b(a(a(x1)))) -> b#(x1)
     b#(b(a(a(x1)))) -> b#(b(x1))
     b#(b(a(a(x1)))) -> a#(b(b(x1)))
     b#(b(a(a(x1)))) -> a#(a(b(b(x1))))
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1)))))
     b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1))))))
    TRS:
     a(x1) -> x1
     b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
    graph:
     b#(b(a(a(x1)))) -> b#(b(x1)) ->
     b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1))))))
     b#(b(a(a(x1)))) -> b#(b(x1)) ->
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1)))))
     b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(a(b(b(x1))))
     b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> a#(b(b(x1)))
     b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(b(x1))
     b#(b(a(a(x1)))) -> b#(b(x1)) -> b#(b(a(a(x1)))) -> b#(x1)
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) ->
     b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1))))))
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) ->
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1)))))
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) ->
     b#(b(a(a(x1)))) -> a#(a(b(b(x1))))
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) ->
     b#(b(a(a(x1)))) -> a#(b(b(x1)))
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) ->
     b#(b(a(a(x1)))) -> b#(b(x1))
     b#(b(a(a(x1)))) -> b#(a(a(b(b(x1))))) -> b#(b(a(a(x1)))) -> b#(x1)
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(b(a(a(b(b(x1))))))
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(a(a(b(b(x1)))))
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(a(b(b(x1))))
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> a#(b(b(x1)))
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(b(x1))
     b#(b(a(a(x1)))) -> b#(x1) -> b#(b(a(a(x1)))) -> b#(x1)
    SCC Processor:
     #sccs: 1
     #rules: 3
     #arcs: 18/36
     DPs:
      b#(b(a(a(x1)))) -> b#(b(x1))
      b#(b(a(a(x1)))) -> b#(x1)
      b#(b(a(a(x1)))) -> b#(a(a(b(b(x1)))))
     TRS:
      a(x1) -> x1
      b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
     Arctic Interpretation Processor:
      dimension: 4
      usable rules:
       a(x1) -> x1
       b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
      interpretation:
       [b#](x0) = [-& -& -& 0 ]x0 + [0],
       
                 [-& -& -& 0 ]     [0]
                 [0  0  -& 0 ]     [0]
       [b](x0) = [0  -& -& 0 ]x0 + [0]
                 [0  -& 0  0 ]     [0],
       
                 [0  -& 0  0 ]     [0]
                 [0  0  0  0 ]     [1]
       [a](x0) = [1  0  0  0 ]x0 + [0]
                 [0  -& -& 0 ]     [0]
      orientation:
       b#(b(a(a(x1)))) = [1 0 1 1]x1 + [1] >= [0  -& 0  0 ]x1 + [0] = b#(b(x1))
       
       b#(b(a(a(x1)))) = [1 0 1 1]x1 + [1] >= [-& -& -& 0 ]x1 + [0] = b#(x1)
       
       b#(b(a(a(x1)))) = [1 0 1 1]x1 + [1] >= [0  -& 0  0 ]x1 + [0] = b#(a(a(b(b(x1)))))
       
               [0  -& 0  0 ]     [0]           
               [0  0  0  0 ]     [1]           
       a(x1) = [1  0  0  0 ]x1 + [0] >= x1 = x1
               [0  -& -& 0 ]     [0]           
       
                        [1 0 1 1]     [1]    [1 0 1 1]     [1]                       
                        [1 0 1 1]     [1]    [1 0 1 1]     [1]                       
       b(b(a(a(x1)))) = [1 0 1 1]x1 + [1] >= [1 0 1 1]x1 + [1] = a(b(a(a(b(b(x1))))))
                        [1 0 1 1]     [1]    [1 0 1 1]     [1]                       
      problem:
       DPs:
        
       TRS:
        a(x1) -> x1
        b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
      Qed