YES

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

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