YES

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

Proof:
 String Reversal Processor:
  a(x1) -> x1
  a(a(x1)) -> c(b(x1))
  b(b(c(x1))) -> a(b(a(a(x1))))
  DP Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    b#(b(c(x1))) -> a#(x1)
    b#(b(c(x1))) -> a#(a(x1))
    b#(b(c(x1))) -> b#(a(a(x1)))
    b#(b(c(x1))) -> a#(b(a(a(x1))))
   TRS:
    a(x1) -> x1
    a(a(x1)) -> c(b(x1))
    b(b(c(x1))) -> a(b(a(a(x1))))
   TDG Processor:
    DPs:
     a#(a(x1)) -> b#(x1)
     b#(b(c(x1))) -> a#(x1)
     b#(b(c(x1))) -> a#(a(x1))
     b#(b(c(x1))) -> b#(a(a(x1)))
     b#(b(c(x1))) -> a#(b(a(a(x1))))
    TRS:
     a(x1) -> x1
     a(a(x1)) -> c(b(x1))
     b(b(c(x1))) -> a(b(a(a(x1))))
    graph:
     b#(b(c(x1))) -> b#(a(a(x1))) -> b#(b(c(x1))) -> a#(b(a(a(x1))))
     b#(b(c(x1))) -> b#(a(a(x1))) -> b#(b(c(x1))) -> b#(a(a(x1)))
     b#(b(c(x1))) -> b#(a(a(x1))) -> b#(b(c(x1))) -> a#(a(x1))
     b#(b(c(x1))) -> b#(a(a(x1))) -> b#(b(c(x1))) -> a#(x1)
     b#(b(c(x1))) -> a#(b(a(a(x1)))) -> a#(a(x1)) -> b#(x1)
     b#(b(c(x1))) -> a#(a(x1)) -> a#(a(x1)) -> b#(x1)
     b#(b(c(x1))) -> a#(x1) -> a#(a(x1)) -> b#(x1)
     a#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> a#(b(a(a(x1))))
     a#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> b#(a(a(x1)))
     a#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> a#(a(x1))
     a#(a(x1)) -> b#(x1) -> b#(b(c(x1))) -> a#(x1)
    Arctic Interpretation Processor:
     dimension: 4
     interpretation:
      [b#](x0) = [0  0  -& 0 ]x0 + [0],
      
      [a#](x0) = [-& 1  0  0 ]x0 + [1],
      
                [0  0  0  0 ]     [-&]
                [0  -& -& 0 ]     [0 ]
      [b](x0) = [0  0  -& 0 ]x0 + [0 ]
                [0  0  -& 0 ]     [0 ],
      
                [0 0 0 0]     [0 ]
                [0 0 0 0]     [1 ]
      [c](x0) = [1 1 1 1]x0 + [-&]
                [0 0 0 0]     [0 ],
      
                [0  0  -& 0 ]     [0]
                [0  0  0  0 ]     [0]
      [a](x0) = [0  1  0  0 ]x0 + [1]
                [0  0  -& 0 ]     [0]
     orientation:
      a#(a(x1)) = [1 1 1 1]x1 + [1] >= [0  0  -& 0 ]x1 + [0] = b#(x1)
      
      b#(b(c(x1))) = [1 1 1 1]x1 + [1] >= [-& 1  0  0 ]x1 + [1] = a#(x1)
      
      b#(b(c(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a#(a(x1))
      
      b#(b(c(x1))) = [1 1 1 1]x1 + [1] >= [0 1 0 0]x1 + [1] = b#(a(a(x1)))
      
      b#(b(c(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a#(b(a(a(x1))))
      
              [0  0  -& 0 ]     [0]           
              [0  0  0  0 ]     [0]           
      a(x1) = [0  1  0  0 ]x1 + [1] >= x1 = x1
              [0  0  -& 0 ]     [0]           
      
                 [0 0 0 0]     [0]    [0 0 0 0]     [0]           
                 [0 1 0 0]     [1]    [0 0 0 0]     [1]           
      a(a(x1)) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = c(b(x1))
                 [0 0 0 0]     [0]    [0 0 0 0]     [0]           
      
                    [1 1 1 1]     [1]    [1 1 1 1]     [1]                 
                    [1 1 1 1]     [1]    [1 1 1 1]     [1]                 
      b(b(c(x1))) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = a(b(a(a(x1))))
                    [1 1 1 1]     [1]    [1 1 1 1]     [1]                 
     problem:
      DPs:
       b#(b(c(x1))) -> a#(x1)
       b#(b(c(x1))) -> a#(a(x1))
       b#(b(c(x1))) -> b#(a(a(x1)))
       b#(b(c(x1))) -> a#(b(a(a(x1))))
      TRS:
       a(x1) -> x1
       a(a(x1)) -> c(b(x1))
       b(b(c(x1))) -> a(b(a(a(x1))))
     EDG Processor:
      DPs:
       b#(b(c(x1))) -> a#(x1)
       b#(b(c(x1))) -> a#(a(x1))
       b#(b(c(x1))) -> b#(a(a(x1)))
       b#(b(c(x1))) -> a#(b(a(a(x1))))
      TRS:
       a(x1) -> x1
       a(a(x1)) -> c(b(x1))
       b(b(c(x1))) -> a(b(a(a(x1))))
      graph:
       b#(b(c(x1))) -> b#(a(a(x1))) -> b#(b(c(x1))) -> a#(x1)
       b#(b(c(x1))) -> b#(a(a(x1))) -> b#(b(c(x1))) -> a#(a(x1))
       b#(b(c(x1))) -> b#(a(a(x1))) -> b#(b(c(x1))) -> b#(a(a(x1)))
       b#(b(c(x1))) -> b#(a(a(x1))) -> b#(b(c(x1))) -> a#(b(a(a(x1))))
      SCC Processor:
       #sccs: 1
       #rules: 1
       #arcs: 4/16
       DPs:
        b#(b(c(x1))) -> b#(a(a(x1)))
       TRS:
        a(x1) -> x1
        a(a(x1)) -> c(b(x1))
        b(b(c(x1))) -> a(b(a(a(x1))))
       Matrix Interpretation Processor: dim=1
        
        interpretation:
         [b#](x0) = 1/2x0,
         
         [b](x0) = 2x0 + 1/2,
         
         [c](x0) = 1/2x0 + 1,
         
         [a](x0) = x0 + 1
        orientation:
         b#(b(c(x1))) = 1/2x1 + 5/4 >= 1/2x1 + 1 = b#(a(a(x1)))
         
         a(x1) = x1 + 1 >= x1 = x1
         
         a(a(x1)) = x1 + 2 >= x1 + 5/4 = c(b(x1))
         
         b(b(c(x1))) = 2x1 + 11/2 >= 2x1 + 11/2 = a(b(a(a(x1))))
        problem:
         DPs:
          
         TRS:
          a(x1) -> x1
          a(a(x1)) -> c(b(x1))
          b(b(c(x1))) -> a(b(a(a(x1))))
        Qed