YES

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

Proof:
 String Reversal Processor:
  a(a(x1)) -> x1
  b(a(x1)) -> c(x1)
  c(c(x1)) -> a(a(b(c(b(x1)))))
  DP Processor:
   DPs:
    b#(a(x1)) -> c#(x1)
    c#(c(x1)) -> b#(x1)
    c#(c(x1)) -> c#(b(x1))
    c#(c(x1)) -> b#(c(b(x1)))
    c#(c(x1)) -> a#(b(c(b(x1))))
    c#(c(x1)) -> a#(a(b(c(b(x1)))))
   TRS:
    a(a(x1)) -> x1
    b(a(x1)) -> c(x1)
    c(c(x1)) -> a(a(b(c(b(x1)))))
   TDG Processor:
    DPs:
     b#(a(x1)) -> c#(x1)
     c#(c(x1)) -> b#(x1)
     c#(c(x1)) -> c#(b(x1))
     c#(c(x1)) -> b#(c(b(x1)))
     c#(c(x1)) -> a#(b(c(b(x1))))
     c#(c(x1)) -> a#(a(b(c(b(x1)))))
    TRS:
     a(a(x1)) -> x1
     b(a(x1)) -> c(x1)
     c(c(x1)) -> a(a(b(c(b(x1)))))
    graph:
     c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(a(b(c(b(x1)))))
     c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> a#(b(c(b(x1))))
     c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(c(b(x1)))
     c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(x1))
     c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(x1)
     c#(c(x1)) -> b#(c(b(x1))) -> b#(a(x1)) -> c#(x1)
     c#(c(x1)) -> b#(x1) -> b#(a(x1)) -> c#(x1)
     b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> a#(a(b(c(b(x1)))))
     b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> a#(b(c(b(x1))))
     b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(c(b(x1)))
     b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(x1))
     b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1)
    SCC Processor:
     #sccs: 1
     #rules: 4
     #arcs: 12/36
     DPs:
      c#(c(x1)) -> c#(b(x1))
      c#(c(x1)) -> b#(x1)
      b#(a(x1)) -> c#(x1)
      c#(c(x1)) -> b#(c(b(x1)))
     TRS:
      a(a(x1)) -> x1
      b(a(x1)) -> c(x1)
      c(c(x1)) -> a(a(b(c(b(x1)))))
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(a(x1)) -> x1
       b(a(x1)) -> c(x1)
       c(c(x1)) -> a(a(b(c(b(x1)))))
      interpretation:
       [c#](x0) = [1 0]x0,
       
       [b#](x0) = [0 1]x0,
       
                 [0  0 ]     [2]
       [c](x0) = [-& -&]x0 + [0],
       
                 [0  0 ]     [2]
       [b](x0) = [-& 0 ]x0 + [0],
       
                 [-& 0 ]     [0]
       [a](x0) = [0  -&]x0 + [0]
      orientation:
       c#(c(x1)) = [1 1]x1 + [3] >= [1 1]x1 + [3] = c#(b(x1))
       
       c#(c(x1)) = [1 1]x1 + [3] >= [0 1]x1 = b#(x1)
       
       b#(a(x1)) = [1 0]x1 + [1] >= [1 0]x1 = c#(x1)
       
       c#(c(x1)) = [1 1]x1 + [3] >= [0 0]x1 + [2] = b#(c(b(x1)))
       
                       [0]           
       a(a(x1)) = x1 + [0] >= x1 = x1
       
                  [0  0 ]     [2]    [0  0 ]     [2]        
       b(a(x1)) = [0  -&]x1 + [0] >= [-& -&]x1 + [0] = c(x1)
       
                  [0  0 ]     [2]    [0  0 ]     [2]                    
       c(c(x1)) = [-& -&]x1 + [0] >= [-& -&]x1 + [0] = a(a(b(c(b(x1)))))
      problem:
       DPs:
        c#(c(x1)) -> c#(b(x1))
        c#(c(x1)) -> b#(x1)
        b#(a(x1)) -> c#(x1)
       TRS:
        a(a(x1)) -> x1
        b(a(x1)) -> c(x1)
        c(c(x1)) -> a(a(b(c(b(x1)))))
      Restore Modifier:
       DPs:
        c#(c(x1)) -> c#(b(x1))
        c#(c(x1)) -> b#(x1)
        b#(a(x1)) -> c#(x1)
       TRS:
        a(a(x1)) -> x1
        b(a(x1)) -> c(x1)
        c(c(x1)) -> a(a(b(c(b(x1)))))
       EDG Processor:
        DPs:
         c#(c(x1)) -> c#(b(x1))
         c#(c(x1)) -> b#(x1)
         b#(a(x1)) -> c#(x1)
        TRS:
         a(a(x1)) -> x1
         b(a(x1)) -> c(x1)
         c(c(x1)) -> a(a(b(c(b(x1)))))
        graph:
         c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> b#(x1)
         c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(x1))
         c#(c(x1)) -> b#(x1) -> b#(a(x1)) -> c#(x1)
         b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> b#(x1)
         b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(x1))
        Arctic Interpretation Processor:
         dimension: 3
         usable rules:
          a(a(x1)) -> x1
          b(a(x1)) -> c(x1)
          c(c(x1)) -> a(a(b(c(b(x1)))))
         interpretation:
          [c#](x0) = [0  0  -&]x0 + [1],
          
          [b#](x0) = [-& -& 0 ]x0 + [0],
          
                    [0  0  -&]     [0 ]
          [c](x0) = [0  0  1 ]x0 + [1 ]
                    [-& 1  0 ]     [-&],
          
                    [-& -& 0 ]     [0 ]
          [b](x0) = [-& -& 0 ]x0 + [-&]
                    [0  -& -&]     [-&],
          
                    [-& 1  0 ]     [1]
          [a](x0) = [-& -& 0 ]x0 + [0]
                    [0  0  1 ]     [1]
         orientation:
          c#(c(x1)) = [0 0 1]x1 + [1] >= [-& -& 0 ]x1 + [1] = c#(b(x1))
          
          c#(c(x1)) = [0 0 1]x1 + [1] >= [-& -& 0 ]x1 + [0] = b#(x1)
          
          b#(a(x1)) = [0 0 1]x1 + [1] >= [0  0  -&]x1 + [1] = c#(x1)
          
                     [0 0 1]     [1]           
          a(a(x1)) = [0 0 1]x1 + [1] >= x1 = x1
                     [1 1 2]     [2]           
          
                     [0  0  1 ]     [1]    [0  0  -&]     [0 ]        
          b(a(x1)) = [0  0  1 ]x1 + [1] >= [0  0  1 ]x1 + [1 ] = c(x1)
                     [-& 1  0 ]     [1]    [-& 1  0 ]     [-&]        
          
                     [0 0 1]     [1]    [0  -& 1 ]     [1]                    
          c(c(x1)) = [0 2 1]x1 + [1] >= [0  -& 1 ]x1 + [1] = a(a(b(c(b(x1)))))
                     [1 1 2]     [2]    [1  -& 2 ]     [2]                    
         problem:
          DPs:
           c#(c(x1)) -> c#(b(x1))
           b#(a(x1)) -> c#(x1)
          TRS:
           a(a(x1)) -> x1
           b(a(x1)) -> c(x1)
           c(c(x1)) -> a(a(b(c(b(x1)))))
         Restore Modifier:
          DPs:
           c#(c(x1)) -> c#(b(x1))
           b#(a(x1)) -> c#(x1)
          TRS:
           a(a(x1)) -> x1
           b(a(x1)) -> c(x1)
           c(c(x1)) -> a(a(b(c(b(x1)))))
          EDG Processor:
           DPs:
            c#(c(x1)) -> c#(b(x1))
            b#(a(x1)) -> c#(x1)
           TRS:
            a(a(x1)) -> x1
            b(a(x1)) -> c(x1)
            c(c(x1)) -> a(a(b(c(b(x1)))))
           graph:
            c#(c(x1)) -> c#(b(x1)) -> c#(c(x1)) -> c#(b(x1))
            b#(a(x1)) -> c#(x1) -> c#(c(x1)) -> c#(b(x1))
           SCC Processor:
            #sccs: 1
            #rules: 1
            #arcs: 2/4
            DPs:
             c#(c(x1)) -> c#(b(x1))
            TRS:
             a(a(x1)) -> x1
             b(a(x1)) -> c(x1)
             c(c(x1)) -> a(a(b(c(b(x1)))))
            Arctic Interpretation Processor:
             dimension: 3
             usable rules:
              a(a(x1)) -> x1
              b(a(x1)) -> c(x1)
              c(c(x1)) -> a(a(b(c(b(x1)))))
             interpretation:
              [c#](x0) = [0 1 0]x0 + [0],
              
                        [0  1  -&]     [1]
              [c](x0) = [1  -& 0 ]x0 + [1]
                        [-& 0  0 ]     [0],
              
                        [-& -& 0 ]     [0 ]
              [b](x0) = [0  -& -&]x0 + [-&]
                        [0  -& -&]     [0 ],
              
                        [1  0  0 ]     [1]
              [a](x0) = [0  -& -&]x0 + [0]
                        [0  1  -&]     [1]
             orientation:
              c#(c(x1)) = [2 1 1]x1 + [2] >= [1  -& 0 ]x1 + [0] = c#(b(x1))
              
                         [2 1 1]     [2]           
              a(a(x1)) = [1 0 0]x1 + [1] >= x1 = x1
                         [1 0 0]     [1]           
              
                         [0  1  -&]     [1]    [0  1  -&]     [1]        
              b(a(x1)) = [1  0  0 ]x1 + [1] >= [1  -& 0 ]x1 + [1] = c(x1)
                         [1  0  0 ]     [1]    [-& 0  0 ]     [0]        
              
                         [2 1 1]     [2]    [2  -& 1 ]     [2]                    
              c(c(x1)) = [1 2 0]x1 + [2] >= [1  -& 0 ]x1 + [1] = a(a(b(c(b(x1)))))
                         [1 0 0]     [1]    [1  -& 0 ]     [1]                    
             problem:
              DPs:
               
              TRS:
               a(a(x1)) -> x1
               b(a(x1)) -> c(x1)
               c(c(x1)) -> a(a(b(c(b(x1)))))
             Qed