YES

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

Proof:
 String Reversal Processor:
  a(x1) -> x1
  b(a(x1)) -> a(c(b(x1)))
  a(c(c(x1))) -> a(b(a(x1)))
  DP Processor:
   DPs:
    b#(a(x1)) -> b#(x1)
    b#(a(x1)) -> a#(c(b(x1)))
    a#(c(c(x1))) -> a#(x1)
    a#(c(c(x1))) -> b#(a(x1))
    a#(c(c(x1))) -> a#(b(a(x1)))
   TRS:
    a(x1) -> x1
    b(a(x1)) -> a(c(b(x1)))
    a(c(c(x1))) -> a(b(a(x1)))
   TDG Processor:
    DPs:
     b#(a(x1)) -> b#(x1)
     b#(a(x1)) -> a#(c(b(x1)))
     a#(c(c(x1))) -> a#(x1)
     a#(c(c(x1))) -> b#(a(x1))
     a#(c(c(x1))) -> a#(b(a(x1)))
    TRS:
     a(x1) -> x1
     b(a(x1)) -> a(c(b(x1)))
     a(c(c(x1))) -> a(b(a(x1)))
    graph:
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(b(x1)))
     b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
     b#(a(x1)) -> a#(c(b(x1))) -> a#(c(c(x1))) -> a#(b(a(x1)))
     b#(a(x1)) -> a#(c(b(x1))) -> a#(c(c(x1))) -> b#(a(x1))
     b#(a(x1)) -> a#(c(b(x1))) -> a#(c(c(x1))) -> a#(x1)
     a#(c(c(x1))) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(b(x1)))
     a#(c(c(x1))) -> b#(a(x1)) -> b#(a(x1)) -> b#(x1)
     a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> a#(b(a(x1)))
     a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> b#(a(x1))
     a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> a#(x1)
     a#(c(c(x1))) -> a#(x1) -> a#(c(c(x1))) -> a#(b(a(x1)))
     a#(c(c(x1))) -> a#(x1) -> a#(c(c(x1))) -> b#(a(x1))
     a#(c(c(x1))) -> a#(x1) -> a#(c(c(x1))) -> a#(x1)
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> x1
      b(a(x1)) -> a(c(b(x1)))
      a(c(c(x1))) -> a(b(a(x1)))
     interpretation:
      [b#](x0) = [0 0]x0 + [0],
      
      [a#](x0) = [0 0]x0,
      
                [0 0]     [-&]
      [c](x0) = [1 0]x0 + [0 ],
      
                [0  -&]  
      [b](x0) = [0  -&]x0,
      
                [1 0]     [0]
      [a](x0) = [0 0]x0 + [0]
     orientation:
      b#(a(x1)) = [1 0]x1 + [0] >= [0 0]x1 + [0] = b#(x1)
      
      b#(a(x1)) = [1 0]x1 + [0] >= [1  -&]x1 + [0] = a#(c(b(x1)))
      
      a#(c(c(x1))) = [1 1]x1 + [0] >= [0 0]x1 = a#(x1)
      
      a#(c(c(x1))) = [1 1]x1 + [0] >= [1 0]x1 + [0] = b#(a(x1))
      
      a#(c(c(x1))) = [1 1]x1 + [0] >= [1 0]x1 + [0] = a#(b(a(x1)))
      
              [1 0]     [0]           
      a(x1) = [0 0]x1 + [0] >= x1 = x1
      
                 [1 0]     [0]    [1  -&]     [0]              
      b(a(x1)) = [1 0]x1 + [0] >= [1  -&]x1 + [0] = a(c(b(x1)))
      
                    [2 1]     [1]    [2 1]     [1]              
      a(c(c(x1))) = [1 1]x1 + [0] >= [1 0]x1 + [0] = a(b(a(x1)))
     problem:
      DPs:
       b#(a(x1)) -> b#(x1)
       b#(a(x1)) -> a#(c(b(x1)))
       a#(c(c(x1))) -> b#(a(x1))
       a#(c(c(x1))) -> a#(b(a(x1)))
      TRS:
       a(x1) -> x1
       b(a(x1)) -> a(c(b(x1)))
       a(c(c(x1))) -> a(b(a(x1)))
     Restore Modifier:
      DPs:
       b#(a(x1)) -> b#(x1)
       b#(a(x1)) -> a#(c(b(x1)))
       a#(c(c(x1))) -> b#(a(x1))
       a#(c(c(x1))) -> a#(b(a(x1)))
      TRS:
       a(x1) -> x1
       b(a(x1)) -> a(c(b(x1)))
       a(c(c(x1))) -> a(b(a(x1)))
      EDG Processor:
       DPs:
        b#(a(x1)) -> b#(x1)
        b#(a(x1)) -> a#(c(b(x1)))
        a#(c(c(x1))) -> b#(a(x1))
        a#(c(c(x1))) -> a#(b(a(x1)))
       TRS:
        a(x1) -> x1
        b(a(x1)) -> a(c(b(x1)))
        a(c(c(x1))) -> a(b(a(x1)))
       graph:
        b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
        b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(c(b(x1)))
        b#(a(x1)) -> a#(c(b(x1))) -> a#(c(c(x1))) -> b#(a(x1))
        b#(a(x1)) -> a#(c(b(x1))) -> a#(c(c(x1))) -> a#(b(a(x1)))
        a#(c(c(x1))) -> b#(a(x1)) -> b#(a(x1)) -> b#(x1)
        a#(c(c(x1))) -> b#(a(x1)) -> b#(a(x1)) -> a#(c(b(x1)))
        a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> b#(a(x1))
        a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> a#(b(a(x1)))
       Arctic Interpretation Processor:
        dimension: 2
        usable rules:
         a(x1) -> x1
         b(a(x1)) -> a(c(b(x1)))
         a(c(c(x1))) -> a(b(a(x1)))
        interpretation:
         [b#](x0) = [0  -&]x0 + [1],
         
         [a#](x0) = [-& 0 ]x0 + [0],
         
                   [0 2]     [2]
         [c](x0) = [0 0]x0 + [0],
         
                   [-& 0 ]     [0]
         [b](x0) = [-& 0 ]x0 + [0],
         
                   [0 1]     [-&]
         [a](x0) = [0 2]x0 + [2 ]
        orientation:
         b#(a(x1)) = [0 1]x1 + [1] >= [0  -&]x1 + [1] = b#(x1)
         
         b#(a(x1)) = [0 1]x1 + [1] >= [-& 0 ]x1 + [0] = a#(c(b(x1)))
         
         a#(c(c(x1))) = [0 2]x1 + [2] >= [0 1]x1 + [1] = b#(a(x1))
         
         a#(c(c(x1))) = [0 2]x1 + [2] >= [0 2]x1 + [2] = a#(b(a(x1)))
         
                 [0 1]     [-&]           
         a(x1) = [0 2]x1 + [2 ] >= x1 = x1
         
                    [0 2]     [2]    [-& 2 ]     [2]              
         b(a(x1)) = [0 2]x1 + [2] >= [-& 2 ]x1 + [2] = a(c(b(x1)))
         
                       [2 3]     [3]    [1 3]     [3]              
         a(c(c(x1))) = [2 4]x1 + [4] >= [2 4]x1 + [4] = a(b(a(x1)))
        problem:
         DPs:
          b#(a(x1)) -> b#(x1)
          a#(c(c(x1))) -> b#(a(x1))
          a#(c(c(x1))) -> a#(b(a(x1)))
         TRS:
          a(x1) -> x1
          b(a(x1)) -> a(c(b(x1)))
          a(c(c(x1))) -> a(b(a(x1)))
        Restore Modifier:
         DPs:
          b#(a(x1)) -> b#(x1)
          a#(c(c(x1))) -> b#(a(x1))
          a#(c(c(x1))) -> a#(b(a(x1)))
         TRS:
          a(x1) -> x1
          b(a(x1)) -> a(c(b(x1)))
          a(c(c(x1))) -> a(b(a(x1)))
         EDG Processor:
          DPs:
           b#(a(x1)) -> b#(x1)
           a#(c(c(x1))) -> b#(a(x1))
           a#(c(c(x1))) -> a#(b(a(x1)))
          TRS:
           a(x1) -> x1
           b(a(x1)) -> a(c(b(x1)))
           a(c(c(x1))) -> a(b(a(x1)))
          graph:
           b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1)
           a#(c(c(x1))) -> b#(a(x1)) -> b#(a(x1)) -> b#(x1)
           a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> a#(b(a(x1)))
           a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> b#(a(x1))
          CDG Processor:
           DPs:
            b#(a(x1)) -> b#(x1)
            a#(c(c(x1))) -> b#(a(x1))
            a#(c(c(x1))) -> a#(b(a(x1)))
           TRS:
            a(x1) -> x1
            b(a(x1)) -> a(c(b(x1)))
            a(c(c(x1))) -> a(b(a(x1)))
           graph:
            a#(c(c(x1))) -> b#(a(x1)) -> b#(a(x1)) -> b#(x1)
            a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> b#(a(x1))
            a#(c(c(x1))) -> a#(b(a(x1))) -> a#(c(c(x1))) -> a#(b(a(x1)))
           SCC Processor:
            #sccs: 1
            #rules: 1
            #arcs: 3/9
            DPs:
             a#(c(c(x1))) -> a#(b(a(x1)))
            TRS:
             a(x1) -> x1
             b(a(x1)) -> a(c(b(x1)))
             a(c(c(x1))) -> a(b(a(x1)))
            Arctic Interpretation Processor:
             dimension: 2
             usable rules:
              a(x1) -> x1
              b(a(x1)) -> a(c(b(x1)))
              a(c(c(x1))) -> a(b(a(x1)))
             interpretation:
              [a#](x0) = [-& 0 ]x0 + [0],
              
                        [0 0]     [-&]
              [c](x0) = [1 1]x0 + [0 ],
              
                        [0  -&]  
              [b](x0) = [0  -&]x0,
              
                        [1 0]     [0]
              [a](x0) = [0 0]x0 + [0]
             orientation:
              a#(c(c(x1))) = [2 2]x1 + [1] >= [1 0]x1 + [0] = a#(b(a(x1)))
              
                      [1 0]     [0]           
              a(x1) = [0 0]x1 + [0] >= x1 = x1
              
                         [1 0]     [0]    [1  -&]     [0]              
              b(a(x1)) = [1 0]x1 + [0] >= [1  -&]x1 + [0] = a(c(b(x1)))
              
                            [2 2]     [1]    [2 1]     [1]              
              a(c(c(x1))) = [2 2]x1 + [1] >= [1 0]x1 + [0] = a(b(a(x1)))
             problem:
              DPs:
               
              TRS:
               a(x1) -> x1
               b(a(x1)) -> a(c(b(x1)))
               a(c(c(x1))) -> a(b(a(x1)))
             Qed