YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(f(b(),a(x)))) -> a#(a(x))
   a#(a(f(b(),a(x)))) -> a#(a(a(x)))
   a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
   a#(a(x)) -> f#(a(x),b())
   a#(a(x)) -> a#(f(a(x),b()))
   a#(a(x)) -> f#(b(),a(f(a(x),b())))
   f#(a(x),b()) -> f#(b(),a(x))
  TRS:
   a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
   a(a(x)) -> f(b(),a(f(a(x),b())))
   f(a(x),b()) -> f(b(),a(x))
  TDG Processor:
   DPs:
    a#(a(f(b(),a(x)))) -> a#(a(x))
    a#(a(f(b(),a(x)))) -> a#(a(a(x)))
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
    a#(a(x)) -> f#(a(x),b())
    a#(a(x)) -> a#(f(a(x),b()))
    a#(a(x)) -> f#(b(),a(f(a(x),b())))
    f#(a(x),b()) -> f#(b(),a(x))
   TRS:
    a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
    a(a(x)) -> f(b(),a(f(a(x),b())))
    f(a(x),b()) -> f(b(),a(x))
   graph:
    f#(a(x),b()) -> f#(b(),a(x)) ->
    f#(a(x),b()) -> f#(b(),a(x))
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b()) ->
    f#(a(x),b()) -> f#(b(),a(x))
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(x)) -> f#(b(),a(f(a(x),b())))
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(x)) -> a#(f(a(x),b()))
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) -> a#(a(x)) -> f#(a(x),b())
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(f(b(),a(x)))) -> a#(a(a(x)))
    a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
    a#(a(f(b(),a(x)))) -> a#(a(x))
    a#(a(f(b(),a(x)))) -> a#(a(x)) ->
    a#(a(x)) -> f#(b(),a(f(a(x),b())))
    a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(x)) -> a#(f(a(x),b()))
    a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(x)) -> f#(a(x),b())
    a#(a(f(b(),a(x)))) -> a#(a(x)) ->
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
    a#(a(f(b(),a(x)))) -> a#(a(x)) ->
    a#(a(f(b(),a(x)))) -> a#(a(a(x)))
    a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(f(b(),a(x)))) -> a#(a(x))
    a#(a(x)) -> f#(a(x),b()) -> f#(a(x),b()) -> f#(b(),a(x))
    a#(a(x)) -> f#(b(),a(f(a(x),b()))) -> f#(a(x),b()) -> f#(b(),a(x))
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(x)) -> f#(b(),a(f(a(x),b())))
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(x)) -> a#(f(a(x),b()))
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(x)) -> f#(a(x),b())
    a#(a(x)) -> a#(f(a(x),b())) ->
    a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(f(b(),a(x)))) -> a#(a(a(x)))
    a#(a(x)) -> a#(f(a(x),b())) -> a#(a(f(b(),a(x)))) -> a#(a(x))
   EDG Processor:
    DPs:
     a#(a(f(b(),a(x)))) -> a#(a(x))
     a#(a(f(b(),a(x)))) -> a#(a(a(x)))
     a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
     a#(a(x)) -> f#(a(x),b())
     a#(a(x)) -> a#(f(a(x),b()))
     a#(a(x)) -> f#(b(),a(f(a(x),b())))
     f#(a(x),b()) -> f#(b(),a(x))
    TRS:
     a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
     a(a(x)) -> f(b(),a(f(a(x),b())))
     f(a(x),b()) -> f(b(),a(x))
    graph:
     a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b()) ->
     f#(a(x),b()) -> f#(b(),a(x))
     a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
     a#(a(f(b(),a(x)))) -> a#(a(x))
     a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
     a#(a(f(b(),a(x)))) -> a#(a(a(x)))
     a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
     a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
     a#(a(f(b(),a(x)))) -> a#(a(a(x))) -> a#(a(x)) -> f#(a(x),b())
     a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
     a#(a(x)) -> a#(f(a(x),b()))
     a#(a(f(b(),a(x)))) -> a#(a(a(x))) ->
     a#(a(x)) -> f#(b(),a(f(a(x),b())))
     a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(f(b(),a(x)))) -> a#(a(x))
     a#(a(f(b(),a(x)))) -> a#(a(x)) ->
     a#(a(f(b(),a(x)))) -> a#(a(a(x)))
     a#(a(f(b(),a(x)))) -> a#(a(x)) ->
     a#(a(f(b(),a(x)))) -> f#(a(a(a(x))),b())
     a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(x)) -> f#(a(x),b())
     a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(x)) -> a#(f(a(x),b()))
     a#(a(f(b(),a(x)))) -> a#(a(x)) -> a#(a(x)) -> f#(b(),a(f(a(x),b())))
     a#(a(x)) -> f#(a(x),b()) -> f#(a(x),b()) -> f#(b(),a(x))
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 14/49
     DPs:
      a#(a(f(b(),a(x)))) -> a#(a(a(x)))
      a#(a(f(b(),a(x)))) -> a#(a(x))
     TRS:
      a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
      a(a(x)) -> f(b(),a(f(a(x),b())))
      f(a(x),b()) -> f(b(),a(x))
     Matrix Interpretation Processor: dim=4
      
      interpretation:
       [a#](x0) = [1 0 0 0]x0 + [1],
       
                     [0 0 0 0]     [0 0 0 0]     [0]
                     [0 0 0 1]     [0 0 0 1]     [1]
       [f](x0, x1) = [1 0 0 0]x0 + [1 0 0 0]x1 + [0]
                     [0 1 0 0]     [0 1 0 0]     [0],
       
                 [0 1 0 1]     [0]
                 [0 1 1 0]     [0]
       [a](x0) = [0 1 0 1]x0 + [1]
                 [0 0 1 1]     [0],
       
             [0]
             [0]
       [b] = [0]
             [0]
      orientation:
       a#(a(f(b(),a(x)))) = [0 1 2 1]x + [2] >= [0 1 2 1]x + [1] = a#(a(a(x)))
       
       a#(a(f(b(),a(x)))) = [0 1 2 1]x + [2] >= [0 1 0 1]x + [1] = a#(a(x))
       
                           [0 3 2 3]    [1]    [0 0 0 0]    [0]                    
                           [0 2 3 3]    [3]    [0 2 3 3]    [3]                    
       a(a(f(b(),a(x)))) = [0 3 2 3]x + [2] >= [0 3 2 3]x + [2] = f(a(a(a(x))),b())
                           [0 3 3 2]    [2]    [0 3 3 2]    [2]                    
       
                 [0 1 2 1]    [0]    [0 0 0 0]    [0]                        
                 [0 2 1 1]    [1]    [0 2 1 1]    [1]                        
       a(a(x)) = [0 1 2 1]x + [1] >= [0 1 2 1]x + [1] = f(b(),a(f(a(x),b())))
                 [0 1 1 2]    [1]    [0 1 1 2]    [1]                        
       
                     [0 0 0 0]    [0]    [0 0 0 0]    [0]              
                     [0 0 1 1]    [1]    [0 0 1 1]    [1]              
       f(a(x),b()) = [0 1 0 1]x + [0] >= [0 1 0 1]x + [0] = f(b(),a(x))
                     [0 1 1 0]    [0]    [0 1 1 0]    [0]              
      problem:
       DPs:
        
       TRS:
        a(a(f(b(),a(x)))) -> f(a(a(a(x))),b())
        a(a(x)) -> f(b(),a(f(a(x),b())))
        f(a(x),b()) -> f(b(),a(x))
      Qed