YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> c#(c(x1))
   a#(a(x1)) -> a#(b(c(c(x1))))
   c#(b(x1)) -> a#(x1)
  TRS:
   a(x1) -> x1
   a(a(x1)) -> a(b(c(c(x1))))
   c(x1) -> b(x1)
   c(b(x1)) -> a(x1)
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> c#(c(x1))
    a#(a(x1)) -> a#(b(c(c(x1))))
    c#(b(x1)) -> a#(x1)
   TRS:
    a(x1) -> x1
    a(a(x1)) -> a(b(c(c(x1))))
    c(x1) -> b(x1)
    c(b(x1)) -> a(x1)
   graph:
    c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(c(c(x1))))
    c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> c#(c(x1))
    c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> c#(c(x1)) -> c#(b(x1)) -> a#(x1)
    a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(x1)
    a#(a(x1)) -> a#(b(c(c(x1)))) -> a#(a(x1)) -> a#(b(c(c(x1))))
    a#(a(x1)) -> a#(b(c(c(x1)))) -> a#(a(x1)) -> c#(c(x1))
    a#(a(x1)) -> a#(b(c(c(x1)))) -> a#(a(x1)) -> c#(x1)
   EDG Processor:
    DPs:
     a#(a(x1)) -> c#(x1)
     a#(a(x1)) -> c#(c(x1))
     a#(a(x1)) -> a#(b(c(c(x1))))
     c#(b(x1)) -> a#(x1)
    TRS:
     a(x1) -> x1
     a(a(x1)) -> a(b(c(c(x1))))
     c(x1) -> b(x1)
     c(b(x1)) -> a(x1)
    graph:
     c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> c#(x1)
     c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> c#(c(x1))
     c#(b(x1)) -> a#(x1) -> a#(a(x1)) -> a#(b(c(c(x1))))
     a#(a(x1)) -> c#(c(x1)) -> c#(b(x1)) -> a#(x1)
     a#(a(x1)) -> c#(x1) -> c#(b(x1)) -> a#(x1)
    SCC Processor:
     #sccs: 1
     #rules: 3
     #arcs: 5/16
     DPs:
      c#(b(x1)) -> a#(x1)
      a#(a(x1)) -> c#(c(x1))
      a#(a(x1)) -> c#(x1)
     TRS:
      a(x1) -> x1
      a(a(x1)) -> a(b(c(c(x1))))
      c(x1) -> b(x1)
      c(b(x1)) -> a(x1)
     Arctic Interpretation Processor:
      dimension: 2
      usable rules:
       a(x1) -> x1
       a(a(x1)) -> a(b(c(c(x1))))
       c(x1) -> b(x1)
       c(b(x1)) -> a(x1)
      interpretation:
       [c#](x0) = [2 0]x0 + [0],
       
       [a#](x0) = [2 3]x0,
       
                 [0  2 ]     [-&]
       [b](x0) = [-& 0 ]x0 + [1 ],
       
                 [0 2]     [-&]
       [c](x0) = [0 0]x0 + [2 ],
       
                 [0 2]     [0]
       [a](x0) = [0 2]x0 + [2]
      orientation:
       c#(b(x1)) = [2 4]x1 + [1] >= [2 3]x1 = a#(x1)
       
       a#(a(x1)) = [3 5]x1 + [5] >= [2 4]x1 + [2] = c#(c(x1))
       
       a#(a(x1)) = [3 5]x1 + [5] >= [2 0]x1 + [0] = c#(x1)
       
               [0 2]     [0]           
       a(x1) = [0 2]x1 + [2] >= x1 = x1
       
                  [2 4]     [4]    [2 4]     [4]                 
       a(a(x1)) = [2 4]x1 + [4] >= [2 4]x1 + [4] = a(b(c(c(x1))))
       
               [0 2]     [-&]    [0  2 ]     [-&]        
       c(x1) = [0 0]x1 + [2 ] >= [-& 0 ]x1 + [1 ] = b(x1)
       
                  [0 2]     [3]    [0 2]     [0]        
       c(b(x1)) = [0 2]x1 + [2] >= [0 2]x1 + [2] = a(x1)
      problem:
       DPs:
        c#(b(x1)) -> a#(x1)
       TRS:
        a(x1) -> x1
        a(a(x1)) -> a(b(c(c(x1))))
        c(x1) -> b(x1)
        c(b(x1)) -> a(x1)
      Restore Modifier:
       DPs:
        c#(b(x1)) -> a#(x1)
       TRS:
        a(x1) -> x1
        a(a(x1)) -> a(b(c(c(x1))))
        c(x1) -> b(x1)
        c(b(x1)) -> a(x1)
       EDG Processor:
        DPs:
         c#(b(x1)) -> a#(x1)
        TRS:
         a(x1) -> x1
         a(a(x1)) -> a(b(c(c(x1))))
         c(x1) -> b(x1)
         c(b(x1)) -> a(x1)
        graph:
         
        Qed