YES

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

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