YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> c#(a(x1))
   a#(b(x1)) -> a#(c(a(x1)))
   a#(b(x1)) -> c#(a(c(a(x1))))
   a#(b(x1)) -> b#(c(a(c(a(x1)))))
   a#(b(x1)) -> b#(b(c(a(c(a(x1))))))
  TRS:
   a(x1) -> x1
   a(b(x1)) -> b(b(c(a(c(a(x1))))))
   b(x1) -> x1
   c(c(x1)) -> x1
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> c#(a(x1))
    a#(b(x1)) -> a#(c(a(x1)))
    a#(b(x1)) -> c#(a(c(a(x1))))
    a#(b(x1)) -> b#(c(a(c(a(x1)))))
    a#(b(x1)) -> b#(b(c(a(c(a(x1))))))
   TRS:
    a(x1) -> x1
    a(b(x1)) -> b(b(c(a(c(a(x1))))))
    b(x1) -> x1
    c(c(x1)) -> x1
   graph:
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> b#(b(c(a(c(a(x1))))))
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> b#(c(a(c(a(x1)))))
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> c#(a(c(a(x1))))
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> a#(c(a(x1)))
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> c#(a(x1))
    a#(b(x1)) -> a#(c(a(x1))) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(b(c(a(c(a(x1))))))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> b#(c(a(c(a(x1)))))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> c#(a(c(a(x1))))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(c(a(x1)))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> c#(a(x1))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 12/36
    DPs:
     a#(b(x1)) -> a#(c(a(x1)))
     a#(b(x1)) -> a#(x1)
    TRS:
     a(x1) -> x1
     a(b(x1)) -> b(b(c(a(c(a(x1))))))
     b(x1) -> x1
     c(c(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> x1
      a(b(x1)) -> b(b(c(a(c(a(x1))))))
      b(x1) -> x1
      c(c(x1)) -> x1
     interpretation:
      [a#](x0) = [1 0]x0 + [0],
      
                [-& 0 ]     [0]
      [c](x0) = [0  -&]x0 + [0],
      
                [0  1 ]     [2]
      [b](x0) = [-& 0 ]x0 + [1],
      
                     [0]
      [a](x0) = x0 + [0]
     orientation:
      a#(b(x1)) = [1 2]x1 + [3] >= [0 1]x1 + [1] = a#(c(a(x1)))
      
      a#(b(x1)) = [1 2]x1 + [3] >= [1 0]x1 + [0] = a#(x1)
      
                   [0]           
      a(x1) = x1 + [0] >= x1 = x1
      
                 [0  1 ]     [2]    [0  1 ]     [2]                       
      a(b(x1)) = [-& 0 ]x1 + [1] >= [-& 0 ]x1 + [1] = b(b(c(a(c(a(x1))))))
      
              [0  1 ]     [2]           
      b(x1) = [-& 0 ]x1 + [1] >= x1 = x1
      
                      [0]           
      c(c(x1)) = x1 + [0] >= x1 = x1
     problem:
      DPs:
       a#(b(x1)) -> a#(x1)
      TRS:
       a(x1) -> x1
       a(b(x1)) -> b(b(c(a(c(a(x1))))))
       b(x1) -> x1
       c(c(x1)) -> x1
     Restore Modifier:
      DPs:
       a#(b(x1)) -> a#(x1)
      TRS:
       a(x1) -> x1
       a(b(x1)) -> b(b(c(a(c(a(x1))))))
       b(x1) -> x1
       c(c(x1)) -> x1
      EDG Processor:
       DPs:
        a#(b(x1)) -> a#(x1)
       TRS:
        a(x1) -> x1
        a(b(x1)) -> b(b(c(a(c(a(x1))))))
        b(x1) -> x1
        c(c(x1)) -> x1
       graph:
        a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
       CDG Processor:
        DPs:
         a#(b(x1)) -> a#(x1)
        TRS:
         a(x1) -> x1
         a(b(x1)) -> b(b(c(a(c(a(x1))))))
         b(x1) -> x1
         c(c(x1)) -> x1
        graph:
         
        Qed