YES

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

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