YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> b#(b(x1))
   a#(a(x1)) -> b#(b(b(x1)))
   a#(a(x1)) -> a#(b(b(b(x1))))
   b#(a(x1)) -> b#(c(x1))
   b#(a(x1)) -> b#(b(c(x1)))
   a#(b(b(c(x1)))) -> b#(x1)
   a#(b(b(c(x1)))) -> a#(b(x1))
   a#(b(b(c(x1)))) -> a#(a(b(x1)))
   a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
  TRS:
   a(a(x1)) -> a(b(b(b(x1))))
   b(a(x1)) -> b(b(c(x1)))
   a(b(b(c(x1)))) -> a(a(a(b(x1))))
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(x1))
    a#(a(x1)) -> b#(b(b(x1)))
    a#(a(x1)) -> a#(b(b(b(x1))))
    b#(a(x1)) -> b#(c(x1))
    b#(a(x1)) -> b#(b(c(x1)))
    a#(b(b(c(x1)))) -> b#(x1)
    a#(b(b(c(x1)))) -> a#(b(x1))
    a#(b(b(c(x1)))) -> a#(a(b(x1)))
    a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
   TRS:
    a(a(x1)) -> a(b(b(b(x1))))
    b(a(x1)) -> b(b(c(x1)))
    a(b(b(c(x1)))) -> a(a(a(b(x1))))
   graph:
    b#(a(x1)) -> b#(c(x1)) -> b#(a(x1)) -> b#(b(c(x1)))
    b#(a(x1)) -> b#(c(x1)) -> b#(a(x1)) -> b#(c(x1))
    b#(a(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> b#(b(c(x1)))
    b#(a(x1)) -> b#(b(c(x1))) -> b#(a(x1)) -> b#(c(x1))
    a#(b(b(c(x1)))) -> b#(x1) -> b#(a(x1)) -> b#(b(c(x1)))
    a#(b(b(c(x1)))) -> b#(x1) -> b#(a(x1)) -> b#(c(x1))
    a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
    a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(b(x1)))
    a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1))
    a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> b#(x1)
    a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(b(b(x1))))
    a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(b(x1)))
    a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(x1))
    a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
    a#(b(b(c(x1)))) -> a#(a(b(x1))) ->
    a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
    a#(b(b(c(x1)))) -> a#(a(b(x1))) ->
    a#(b(b(c(x1)))) -> a#(a(b(x1)))
    a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(b(x1))
    a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> b#(x1)
    a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(b(b(x1))))
    a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(b(x1)))
    a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(x1))
    a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(x1)
    a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
    a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
    a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
    a#(b(b(c(x1)))) -> a#(a(b(x1)))
    a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
    a#(b(b(c(x1)))) -> a#(b(x1))
    a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
    a#(b(b(c(x1)))) -> b#(x1)
    a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
    a#(a(x1)) -> a#(b(b(b(x1))))
    a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
    a#(a(x1)) -> b#(b(b(x1)))
    a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(b(x1))
    a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> b#(b(c(x1)))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> b#(c(x1))
    a#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(c(x1)))
    a#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(c(x1))
    a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(c(x1)))
    a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(c(x1))
    a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
    a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(b(x1)))
    a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(b(x1))
    a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> b#(x1)
    a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> a#(b(b(b(x1))))
    a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(b(b(x1)))
    a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(b(x1))
    a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(x1)
   EDG Processor:
    DPs:
     a#(a(x1)) -> b#(x1)
     a#(a(x1)) -> b#(b(x1))
     a#(a(x1)) -> b#(b(b(x1)))
     a#(a(x1)) -> a#(b(b(b(x1))))
     b#(a(x1)) -> b#(c(x1))
     b#(a(x1)) -> b#(b(c(x1)))
     a#(b(b(c(x1)))) -> b#(x1)
     a#(b(b(c(x1)))) -> a#(b(x1))
     a#(b(b(c(x1)))) -> a#(a(b(x1)))
     a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
    TRS:
     a(a(x1)) -> a(b(b(b(x1))))
     b(a(x1)) -> b(b(c(x1)))
     a(b(b(c(x1)))) -> a(a(a(b(x1))))
    graph:
     a#(b(b(c(x1)))) -> b#(x1) -> b#(a(x1)) -> b#(c(x1))
     a#(b(b(c(x1)))) -> b#(x1) -> b#(a(x1)) -> b#(b(c(x1)))
     a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
     a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(x1))
     a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(b(x1)))
     a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(b(b(x1))))
     a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> b#(x1)
     a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1))
     a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(b(x1)))
     a#(b(b(c(x1)))) -> a#(b(x1)) ->
     a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
     a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(x1)
     a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(x1))
     a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(b(x1)))
     a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> a#(b(b(b(x1))))
     a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> b#(x1)
     a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> a#(b(x1))
     a#(b(b(c(x1)))) -> a#(a(b(x1))) ->
     a#(b(b(c(x1)))) -> a#(a(b(x1)))
     a#(b(b(c(x1)))) -> a#(a(b(x1))) ->
     a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
     a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(x1)
     a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(b(x1))
     a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
     a#(a(x1)) -> b#(b(b(x1)))
     a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
     a#(a(x1)) -> a#(b(b(b(x1))))
     a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
     a#(b(b(c(x1)))) -> b#(x1)
     a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
     a#(b(b(c(x1)))) -> a#(b(x1))
     a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
     a#(b(b(c(x1)))) -> a#(a(b(x1)))
     a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
     a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
     a#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> b#(c(x1))
     a#(a(x1)) -> b#(b(b(x1))) -> b#(a(x1)) -> b#(b(c(x1)))
     a#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(c(x1))
     a#(a(x1)) -> b#(b(x1)) -> b#(a(x1)) -> b#(b(c(x1)))
     a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(c(x1))
     a#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(b(c(x1)))
     a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(x1)
     a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(b(x1))
     a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> b#(b(b(x1)))
     a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(a(x1)) -> a#(b(b(b(x1))))
     a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> b#(x1)
     a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(b(x1))
     a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(b(x1)))
     a#(a(x1)) -> a#(b(b(b(x1)))) -> a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
    CDG Processor:
     DPs:
      a#(a(x1)) -> b#(x1)
      a#(a(x1)) -> b#(b(x1))
      a#(a(x1)) -> b#(b(b(x1)))
      a#(a(x1)) -> a#(b(b(b(x1))))
      b#(a(x1)) -> b#(c(x1))
      b#(a(x1)) -> b#(b(c(x1)))
      a#(b(b(c(x1)))) -> b#(x1)
      a#(b(b(c(x1)))) -> a#(b(x1))
      a#(b(b(c(x1)))) -> a#(a(b(x1)))
      a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
     TRS:
      a(a(x1)) -> a(b(b(b(x1))))
      b(a(x1)) -> b(b(c(x1)))
      a(b(b(c(x1)))) -> a(a(a(b(x1))))
     graph:
      a#(b(b(c(x1)))) -> a#(b(x1)) ->
      a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
      a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(a(b(x1)))
      a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> a#(b(x1))
      a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(b(b(c(x1)))) -> b#(x1)
      a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> a#(b(b(b(x1))))
      a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(b(x1)))
      a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(b(x1))
      a#(b(b(c(x1)))) -> a#(b(x1)) -> a#(a(x1)) -> b#(x1)
      a#(b(b(c(x1)))) -> a#(a(b(x1))) ->
      a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
      a#(b(b(c(x1)))) -> a#(a(b(x1))) ->
      a#(b(b(c(x1)))) -> a#(a(b(x1)))
      a#(b(b(c(x1)))) -> a#(a(b(x1))) ->
      a#(b(b(c(x1)))) -> a#(b(x1))
      a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(b(b(c(x1)))) -> b#(x1)
      a#(b(b(c(x1)))) -> a#(a(b(x1))) ->
      a#(a(x1)) -> a#(b(b(b(x1))))
      a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(b(x1)))
      a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(b(x1))
      a#(b(b(c(x1)))) -> a#(a(b(x1))) -> a#(a(x1)) -> b#(x1)
      a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
      a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
      a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
      a#(b(b(c(x1)))) -> a#(a(b(x1)))
      a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
      a#(b(b(c(x1)))) -> a#(b(x1))
      a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
      a#(b(b(c(x1)))) -> b#(x1)
      a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
      a#(a(x1)) -> a#(b(b(b(x1))))
      a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
      a#(a(x1)) -> b#(b(b(x1)))
      a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) ->
      a#(a(x1)) -> b#(b(x1))
      a#(b(b(c(x1)))) -> a#(a(a(b(x1)))) -> a#(a(x1)) -> b#(x1)
     SCC Processor:
      #sccs: 1
      #rules: 3
      #arcs: 24/100
      DPs:
       a#(b(b(c(x1)))) -> a#(b(x1))
       a#(b(b(c(x1)))) -> a#(a(b(x1)))
       a#(b(b(c(x1)))) -> a#(a(a(b(x1))))
      TRS:
       a(a(x1)) -> a(b(b(b(x1))))
       b(a(x1)) -> b(b(c(x1)))
       a(b(b(c(x1)))) -> a(a(a(b(x1))))
      Matrix Interpretation Processor: dim=3
       
       interpretation:
        [a#](x0) = [1 0 0]x0,
        
                  [0 0 0]     [0]
        [c](x0) = [0 0 2]x0 + [1]
                  [1 1 1]     [2],
        
                  [0 1 0]  
        [b](x0) = [0 0 1]x0
                  [0 0 0]  ,
        
                  [0 0 0]     [0]
        [a](x0) = [2 1 2]x0 + [2]
                  [0 0 0]     [0]
       orientation:
        a#(b(b(c(x1)))) = [1 1 1]x1 + [2] >= [0 1 0]x1 = a#(b(x1))
        
        a#(b(b(c(x1)))) = [1 1 1]x1 + [2] >= [0] = a#(a(b(x1)))
        
        a#(b(b(c(x1)))) = [1 1 1]x1 + [2] >= [0] = a#(a(a(b(x1))))
        
                   [0 0 0]     [0]    [0]                 
        a(a(x1)) = [2 1 2]x1 + [4] >= [2] = a(b(b(b(x1))))
                   [0 0 0]     [0]    [0]                 
        
                   [2 1 2]     [2]    [1 1 1]     [2]              
        b(a(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = b(b(c(x1)))
                   [0 0 0]     [0]    [0 0 0]     [0]              
        
                         [0 0 0]     [0]    [0 0 0]     [0]                 
        a(b(b(c(x1)))) = [2 2 2]x1 + [6] >= [0 2 1]x1 + [6] = a(a(a(b(x1))))
                         [0 0 0]     [0]    [0 0 0]     [0]                 
       problem:
        DPs:
         
        TRS:
         a(a(x1)) -> a(b(b(b(x1))))
         b(a(x1)) -> b(b(c(x1)))
         a(b(b(c(x1)))) -> a(a(a(b(x1))))
       Qed