NO

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

Proof:
 DP Processor:
  DPs:
   a#(x1) -> c#(x1)
   a#(b(b(x1))) -> a#(x1)
   a#(b(b(x1))) -> a#(a(x1))
   a#(b(b(x1))) -> a#(a(a(x1)))
   a#(b(b(x1))) -> c#(a(a(a(x1))))
  TRS:
   a(x1) -> b(c(x1))
   a(b(b(x1))) -> c(a(a(a(x1))))
   c(c(x1)) -> x1
  TDG Processor:
   DPs:
    a#(x1) -> c#(x1)
    a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> a#(a(x1))
    a#(b(b(x1))) -> a#(a(a(x1)))
    a#(b(b(x1))) -> c#(a(a(a(x1))))
   TRS:
    a(x1) -> b(c(x1))
    a(b(b(x1))) -> c(a(a(a(x1))))
    c(c(x1)) -> x1
   graph:
    a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> c#(a(a(a(x1))))
    a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1)))
    a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1))
    a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> a#(a(a(x1))) -> a#(x1) -> c#(x1)
    a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> c#(a(a(a(x1))))
    a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(a(x1)))
    a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(x1))
    a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> a#(a(x1)) -> a#(x1) -> c#(x1)
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> c#(a(a(a(x1))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(a(a(x1)))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(a(x1))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> a#(x1) -> a#(x1) -> c#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 3
    #arcs: 15/25
    DPs:
     a#(b(b(x1))) -> a#(a(a(x1)))
     a#(b(b(x1))) -> a#(x1)
     a#(b(b(x1))) -> a#(a(x1))
    TRS:
     a(x1) -> b(c(x1))
     a(b(b(x1))) -> c(a(a(a(x1))))
     c(c(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> b(c(x1))
      a(b(b(x1))) -> c(a(a(a(x1))))
      c(c(x1)) -> x1
     interpretation:
      [a#](x0) = [-& 0 ]x0 + [0],
      
                [-& 2 ]     [2]
      [b](x0) = [0  0 ]x0 + [0],
      
                [0  0 ]     [0]
      [c](x0) = [-& 0 ]x0 + [0],
      
                [0 2]     [2]
      [a](x0) = [0 0]x0 + [0]
     orientation:
      a#(b(b(x1))) = [0 2]x1 + [2] >= [0 2]x1 + [2] = a#(a(a(x1)))
      
      a#(b(b(x1))) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a#(x1)
      
      a#(b(b(x1))) = [0 2]x1 + [2] >= [0 0]x1 + [0] = a#(a(x1))
      
              [0 2]     [2]    [-& 2 ]     [2]           
      a(x1) = [0 0]x1 + [0] >= [0  0 ]x1 + [0] = b(c(x1))
      
                    [2 4]     [4]    [2 4]     [4]                 
      a(b(b(x1))) = [2 2]x1 + [2] >= [2 2]x1 + [2] = c(a(a(a(x1))))
      
                 [0  0 ]     [0]           
      c(c(x1)) = [-& 0 ]x1 + [0] >= x1 = x1
     problem:
      DPs:
       a#(b(b(x1))) -> a#(a(a(x1)))
       a#(b(b(x1))) -> a#(a(x1))
      TRS:
       a(x1) -> b(c(x1))
       a(b(b(x1))) -> c(a(a(a(x1))))
       c(c(x1)) -> x1
     Restore Modifier:
      DPs:
       a#(b(b(x1))) -> a#(a(a(x1)))
       a#(b(b(x1))) -> a#(a(x1))
      TRS:
       a(x1) -> b(c(x1))
       a(b(b(x1))) -> c(a(a(a(x1))))
       c(c(x1)) -> x1
      EDG Processor:
       DPs:
        a#(b(b(x1))) -> a#(a(a(x1)))
        a#(b(b(x1))) -> a#(a(x1))
       TRS:
        a(x1) -> b(c(x1))
        a(b(b(x1))) -> c(a(a(a(x1))))
        c(c(x1)) -> x1
       graph:
        a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1))
        a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1)))
        a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(x1))
        a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(a(x1)))
       Arctic Interpretation Processor:
        dimension: 2
        usable rules:
         a(x1) -> b(c(x1))
         a(b(b(x1))) -> c(a(a(a(x1))))
         c(c(x1)) -> x1
        interpretation:
         [a#](x0) = [0  -&]x0 + [0],
         
                   [0 2]     [0]
         [b](x0) = [0 2]x0 + [2],
         
                   [0  0 ]     [0 ]
         [c](x0) = [-& 0 ]x0 + [-&],
         
                   [0 2]     [0]
         [a](x0) = [0 2]x0 + [2]
        orientation:
         a#(b(b(x1))) = [2 4]x1 + [4] >= [2 4]x1 + [4] = a#(a(a(x1)))
         
         a#(b(b(x1))) = [2 4]x1 + [4] >= [0 2]x1 + [0] = a#(a(x1))
         
                 [0 2]     [0]    [0 2]     [0]           
         a(x1) = [0 2]x1 + [2] >= [0 2]x1 + [2] = b(c(x1))
         
                       [4 6]     [6]    [4 6]     [6]                 
         a(b(b(x1))) = [4 6]x1 + [6] >= [4 6]x1 + [6] = c(a(a(a(x1))))
         
                    [0  0 ]     [0 ]           
         c(c(x1)) = [-& 0 ]x1 + [-&] >= x1 = x1
        problem:
         DPs:
          a#(b(b(x1))) -> a#(a(a(x1)))
         TRS:
          a(x1) -> b(c(x1))
          a(b(b(x1))) -> c(a(a(a(x1))))
          c(c(x1)) -> x1
        Restore Modifier:
         DPs:
          a#(b(b(x1))) -> a#(a(a(x1)))
         TRS:
          a(x1) -> b(c(x1))
          a(b(b(x1))) -> c(a(a(a(x1))))
          c(c(x1)) -> x1
         EDG Processor:
          DPs:
           a#(b(b(x1))) -> a#(a(a(x1)))
          TRS:
           a(x1) -> b(c(x1))
           a(b(b(x1))) -> c(a(a(a(x1))))
           c(c(x1)) -> x1
          graph:
           a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1)))
          Loop Processor:
           loop length: 17
           terms:
            a(a(a(b(b(c(a(a(c(c(c(c(c(c(c(x1)))))))))))))))
            a(a(c(a(a(a(c(a(a(c(c(c(c(c(c(c(x1))))))))))))))))
            a(a(c(b(c(a(a(c(a(a(c(c(c(c(c(c(c(x1)))))))))))))))))
            a(a(c(b(c(a(b(c(c(a(a(c(c(c(c(c(c(c(x1))))))))))))))))))
            a(a(c(b(c(a(b(c(c(b(c(a(c(c(c(c(c(c(c(x1)))))))))))))))))))
            a(a(c(b(c(a(b(b(c(a(c(c(c(c(c(c(c(x1)))))))))))))))))
            a(b(c(c(b(c(a(b(b(c(a(c(c(c(c(c(c(c(x1))))))))))))))))))
            a(b(c(c(b(c(c(a(a(a(c(a(c(c(c(c(c(c(c(x1)))))))))))))))))))
            a(b(c(c(b(c(c(b(c(a(a(c(a(c(c(c(c(c(c(c(x1))))))))))))))))))))
            a(b(b(c(c(b(c(a(a(c(a(c(c(c(c(c(c(c(x1))))))))))))))))))
            a(b(b(c(c(b(c(a(b(c(c(a(c(c(c(c(c(c(c(x1)))))))))))))))))))
            a(b(b(c(c(b(c(a(b(c(c(b(c(c(c(c(c(c(c(c(x1))))))))))))))))))))
            a(b(b(b(c(a(b(c(c(b(c(c(c(c(c(c(c(c(x1))))))))))))))))))
            c(a(a(a(b(c(a(b(c(c(b(c(c(c(c(c(c(c(c(x1)))))))))))))))))))
            c(a(a(a(b(c(a(b(b(c(c(c(c(c(c(c(c(x1)))))))))))))))))
            c(a(a(a(b(c(c(a(a(a(c(c(c(c(c(c(c(c(x1))))))))))))))))))
            c(a(a(a(b(c(c(b(c(a(a(c(c(c(c(c(c(c(c(x1)))))))))))))))))))
           context: c([])
           substitution:
            x1 -> c(x1)
           Qed