YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(b(x1))) -> a#(x1)
   a#(b(b(x1))) -> a#(a(x1))
   a#(b(b(x1))) -> c#(a(a(x1)))
  TRS:
   a(x1) -> x1
   a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
   c(b(x1)) -> x1
  TDG Processor:
   DPs:
    a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> a#(a(x1))
    a#(b(b(x1))) -> c#(a(a(x1)))
   TRS:
    a(x1) -> x1
    a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
    c(b(x1)) -> x1
   graph:
    a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> c#(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#(x1) -> a#(b(b(x1))) -> c#(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)
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 6/9
    DPs:
     a#(b(b(x1))) -> a#(a(x1))
     a#(b(b(x1))) -> a#(x1)
    TRS:
     a(x1) -> x1
     a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
     c(b(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     usable rules:
      a(x1) -> x1
      a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
      c(b(x1)) -> x1
     interpretation:
      [a#](x0) = [0 0]x0,
      
                [0  -&]  
      [c](x0) = [0  -&]x0,
      
                [0 0]     [-&]
      [b](x0) = [2 0]x0 + [0 ],
      
                [0  -&]     [-&]
      [a](x0) = [2  0 ]x0 + [0 ]
     orientation:
      a#(b(b(x1))) = [2 2]x1 + [0] >= [2 0]x1 + [0] = a#(a(x1))
      
      a#(b(b(x1))) = [2 2]x1 + [0] >= [0 0]x1 = a#(x1)
      
              [0  -&]     [-&]           
      a(x1) = [2  0 ]x1 + [0 ] >= x1 = x1
      
                    [2 0]     [0]    [2  -&]     [0]                       
      a(b(b(x1))) = [4 2]x1 + [2] >= [4  -&]x1 + [2] = b(b(b(c(a(a(x1))))))
      
                 [0 0]             
      c(b(x1)) = [0 0]x1 >= x1 = x1
     problem:
      DPs:
       a#(b(b(x1))) -> a#(a(x1))
      TRS:
       a(x1) -> x1
       a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
       c(b(x1)) -> x1
     Restore Modifier:
      DPs:
       a#(b(b(x1))) -> a#(a(x1))
      TRS:
       a(x1) -> x1
       a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
       c(b(x1)) -> x1
      EDG Processor:
       DPs:
        a#(b(b(x1))) -> a#(a(x1))
       TRS:
        a(x1) -> x1
        a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
        c(b(x1)) -> x1
       graph:
        a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(x1))
       Arctic Interpretation Processor:
        dimension: 2
        usable rules:
         a(x1) -> x1
         a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
         c(b(x1)) -> x1
        interpretation:
         [a#](x0) = [0 2]x0 + [0],
         
                   [-& 0 ]     [0]
         [c](x0) = [-& 0 ]x0 + [2],
         
                   [0 1]     [3]
         [b](x0) = [0 0]x0 + [0],
         
                   [0  1 ]     [3]
         [a](x0) = [-& 0 ]x0 + [0]
        orientation:
         a#(b(b(x1))) = [2 3]x1 + [5] >= [0 2]x1 + [3] = a#(a(x1))
         
                 [0  1 ]     [3]           
         a(x1) = [-& 0 ]x1 + [0] >= x1 = x1
         
                       [1 2]     [4]    [-& 2 ]     [4]                       
         a(b(b(x1))) = [0 1]x1 + [3] >= [-& 1 ]x1 + [3] = b(b(b(c(a(a(x1))))))
         
                    [0 0]     [0]           
         c(b(x1)) = [0 0]x1 + [2] >= x1 = x1
        problem:
         DPs:
          
         TRS:
          a(x1) -> x1
          a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
          c(b(x1)) -> x1
        Qed