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
    interpretation:
     [c#](x0) = [3 0]x0 + [0],
     
     [a#](x0) = [0 3]x0 + [0],
     
               [2  0 ]     [2]
     [c](x0) = [2  -&]x0 + [0],
     
               [-& -&]     [0]
     [b](x0) = [0  0 ]x0 + [1],
     
               [0  -&]     [0]
     [a](x0) = [0  2 ]x0 + [3]
    orientation:
     a#(b(x1)) = [3 3]x1 + [4] >= [3 0]x1 + [0] = c#(x1)
     
     c#(c(x1)) = [5 3]x1 + [5] >= [0 3]x1 + [0] = a#(x1)
     
     c#(c(x1)) = [5 3]x1 + [5] >= [3 2]x1 + [3] = c#(a(x1))
     
     c#(c(x1)) = [5 3]x1 + [5] >= [5 2]x1 + [5] = a#(c(a(x1)))
     
             [0  -&]     [0]           
     a(x1) = [0  2 ]x1 + [3] >= x1 = x1
     
             [0  -&]     [0]    [-& -&]     [0]        
     a(x1) = [0  2 ]x1 + [3] >= [0  0 ]x1 + [1] = b(x1)
     
                [-& -&]     [0]    [-& -&]     [0]           
     a(b(x1)) = [2  2 ]x1 + [3] >= [2  0 ]x1 + [2] = b(c(x1))
     
                [4 2]     [4]    [2 2]     [3]              
     c(c(x1)) = [4 2]x1 + [4] >= [4 2]x1 + [4] = a(c(a(x1)))
    problem:
     DPs:
      a#(b(x1)) -> c#(x1)
      c#(c(x1)) -> 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)))
    EDG Processor:
     DPs:
      a#(b(x1)) -> c#(x1)
      c#(c(x1)) -> 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)) -> 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#(x1)
      a#(b(x1)) -> c#(x1) -> c#(c(x1)) -> a#(c(a(x1)))
     Arctic Interpretation Processor:
      dimension: 2
      interpretation:
       [c#](x0) = [0 2]x0,
       
       [a#](x0) = [1 0]x0,
       
                 [0 2]  
       [c](x0) = [0 1]x0,
       
                 [0  1 ]  
       [b](x0) = [-& -&]x0,
       
                 [1  1 ]  
       [a](x0) = [-& 0 ]x0
      orientation:
       a#(b(x1)) = [1 2]x1 >= [0 2]x1 = c#(x1)
       
       c#(c(x1)) = [2 3]x1 >= [1 0]x1 = a#(x1)
       
       c#(c(x1)) = [2 3]x1 >= [2 3]x1 = a#(c(a(x1)))
       
               [1  1 ]             
       a(x1) = [-& 0 ]x1 >= x1 = x1
       
               [1  1 ]      [0  1 ]          
       a(x1) = [-& 0 ]x1 >= [-& -&]x1 = b(x1)
       
                  [1  2 ]      [1  2 ]             
       a(b(x1)) = [-& -&]x1 >= [-& -&]x1 = b(c(x1))
       
                  [2 3]      [2 3]                
       c(c(x1)) = [1 2]x1 >= [1 1]x1 = 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)))
      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: 2
        interpretation:
         [c#](x0) = [1 0]x0 + [0],
         
         [a#](x0) = [0  -&]x0 + [0],
         
                   [-& 0 ]     [0]
         [c](x0) = [-& 1 ]x0 + [1],
         
                   [1  0 ]     [-&]
         [b](x0) = [-& -&]x0 + [0 ],
         
                   [1  0 ]     [1]
         [a](x0) = [-& 0 ]x0 + [0]
        orientation:
         a#(b(x1)) = [1 0]x1 + [0] >= [1 0]x1 + [0] = c#(x1)
         
         c#(c(x1)) = [-& 1 ]x1 + [1] >= [-& 0 ]x1 + [0] = a#(c(a(x1)))
         
                 [1  0 ]     [1]           
         a(x1) = [-& 0 ]x1 + [0] >= x1 = x1
         
                 [1  0 ]     [1]    [1  0 ]     [-&]        
         a(x1) = [-& 0 ]x1 + [0] >= [-& -&]x1 + [0 ] = b(x1)
         
                    [2  1 ]     [1]    [-& 1 ]     [1]           
         a(b(x1)) = [-& -&]x1 + [0] >= [-& -&]x1 + [0] = b(c(x1))
         
                    [-& 1 ]     [1]    [-& 1 ]     [1]              
         c(c(x1)) = [-& 2 ]x1 + [2] >= [-& 1 ]x1 + [1] = a(c(a(x1)))
        problem:
         DPs:
          a#(b(x1)) -> c#(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)
         TRS:
          a(x1) -> x1
          a(x1) -> b(x1)
          a(b(x1)) -> b(c(x1))
          c(c(x1)) -> a(c(a(x1)))
         graph:
          
         Qed