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) = [0  -&]x0 + [0],
     
               [-& -&]     [0]
     [b](x0) = [0  0 ]x0 + [1],
     
     [a#](x0) = [-& 0 ]x0 + [2],
     
               [0  -&]     [1]
     [a](x0) = [0  1 ]x0 + [1],
     
               [1  0 ]     [2]
     [c](x0) = [1  -&]x0 + [0]
    orientation:
     a#(b(x1)) = [0 0]x1 + [2] >= [0  -&]x1 + [0] = c#(x1)
     
     c#(c(x1)) = [1 0]x1 + [2] >= [-& 0 ]x1 + [2] = a#(x1)
     
     c#(c(x1)) = [1 0]x1 + [2] >= [0  -&]x1 + [1] = c#(a(x1))
     
     c#(c(x1)) = [1 0]x1 + [2] >= [1  -&]x1 + [2] = a#(c(a(x1)))
     
             [0  -&]     [1]           
     a(x1) = [0  1 ]x1 + [1] >= x1 = x1
     
             [0  -&]     [1]    [-& -&]     [0]        
     a(x1) = [0  1 ]x1 + [1] >= [0  0 ]x1 + [1] = b(x1)
     
                [-& -&]     [1]    [-& -&]     [0]           
     a(b(x1)) = [1  1 ]x1 + [2] >= [1  0 ]x1 + [2] = b(c(x1))
     
                [2 1]     [3]    [1 1]     [2]              
     c(c(x1)) = [2 1]x1 + [3] >= [2 1]x1 + [3] = 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)))
    Restore Modifier:
     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
       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 1]x0,
        
                  [0  1 ]     [0 ]
        [b](x0) = [-& -&]x0 + [-&],
        
        [a#](x0) = [0  -&]x0,
        
                  [2  1 ]     [2 ]
        [a](x0) = [-& 0 ]x0 + [-&],
        
                  [-& 3 ]     [0]
        [c](x0) = [0  2 ]x0 + [0]
       orientation:
        a#(b(x1)) = [0 1]x1 + [0] >= [0 1]x1 = c#(x1)
        
        c#(c(x1)) = [1 3]x1 + [1] >= [0  -&]x1 = a#(x1)
        
        c#(c(x1)) = [1 3]x1 + [1] >= [-& 3 ]x1 + [0] = a#(c(a(x1)))
        
                [2  1 ]     [2 ]           
        a(x1) = [-& 0 ]x1 + [-&] >= x1 = x1
        
                [2  1 ]     [2 ]    [0  1 ]     [0 ]        
        a(x1) = [-& 0 ]x1 + [-&] >= [-& -&]x1 + [-&] = b(x1)
        
                   [2  3 ]     [2 ]    [1  3 ]     [1 ]           
        a(b(x1)) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = b(c(x1))
        
                   [3 5]     [3]    [3 5]     [3]              
        c(c(x1)) = [2 4]x1 + [2] >= [2 2]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: 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) = [2 2]x0 + [3],
           
                     [0  0 ]     [0 ]
           [b](x0) = [-& -&]x0 + [-&],
           
           [a#](x0) = [2 0]x0 + [3],
           
                     [2  0 ]     [3 ]
           [a](x0) = [-& 0 ]x0 + [-&],
           
                     [-& 0 ]     [1]
           [c](x0) = [-& 2 ]x0 + [3]
          orientation:
           a#(b(x1)) = [2 2]x1 + [3] >= [2 2]x1 + [3] = c#(x1)
           
           c#(c(x1)) = [-& 4 ]x1 + [5] >= [-& 2 ]x1 + [3] = a#(c(a(x1)))
           
                   [2  0 ]     [3 ]           
           a(x1) = [-& 0 ]x1 + [-&] >= x1 = x1
           
                   [2  0 ]     [3 ]    [0  0 ]     [0 ]        
           a(x1) = [-& 0 ]x1 + [-&] >= [-& -&]x1 + [-&] = b(x1)
           
                      [2  2 ]     [3 ]    [-& 2 ]     [3 ]           
           a(b(x1)) = [-& -&]x1 + [-&] >= [-& -&]x1 + [-&] = b(c(x1))
           
                      [-& 2 ]     [3]    [-& 2 ]     [3]              
           c(c(x1)) = [-& 4 ]x1 + [5] >= [-& 2 ]x1 + [3] = 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)))
          Restore Modifier:
           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:
             
            SCC Processor:
             #sccs: 0
             #rules: 0
             #arcs: 0/1