YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(b(x1))) -> a#(x1)
   a#(b(b(x1))) -> b#(c(a(x1)))
   a#(b(b(x1))) -> a#(b(c(a(x1))))
   a#(b(b(x1))) -> b#(a(b(c(a(x1)))))
   a#(b(b(x1))) -> b#(b(a(b(c(a(x1))))))
  TRS:
   a(x1) -> x1
   a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
   b(c(x1)) -> x1
  TDG Processor:
   DPs:
    a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> b#(c(a(x1)))
    a#(b(b(x1))) -> a#(b(c(a(x1))))
    a#(b(b(x1))) -> b#(a(b(c(a(x1)))))
    a#(b(b(x1))) -> b#(b(a(b(c(a(x1))))))
   TRS:
    a(x1) -> x1
    a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
    b(c(x1)) -> x1
   graph:
    a#(b(b(x1))) -> a#(b(c(a(x1)))) ->
    a#(b(b(x1))) -> b#(b(a(b(c(a(x1))))))
    a#(b(b(x1))) -> a#(b(c(a(x1)))) ->
    a#(b(b(x1))) -> b#(a(b(c(a(x1)))))
    a#(b(b(x1))) -> a#(b(c(a(x1)))) ->
    a#(b(b(x1))) -> a#(b(c(a(x1))))
    a#(b(b(x1))) -> a#(b(c(a(x1)))) -> a#(b(b(x1))) -> b#(c(a(x1)))
    a#(b(b(x1))) -> a#(b(c(a(x1)))) -> a#(b(b(x1))) -> a#(x1)
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(b(a(b(c(a(x1))))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(a(b(c(a(x1)))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(b(c(a(x1))))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> b#(c(a(x1)))
    a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 10/25
    DPs:
     a#(b(b(x1))) -> a#(b(c(a(x1))))
     a#(b(b(x1))) -> a#(x1)
    TRS:
     a(x1) -> x1
     a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
     b(c(x1)) -> x1
    Arctic Interpretation Processor:
     dimension: 2
     interpretation:
      [a#](x0) = [0 0]x0 + [0],
      
                [-& -&]     [0]
      [c](x0) = [0  0 ]x0 + [3],
      
                [-& 0 ]     [3]
      [b](x0) = [1  0 ]x0 + [3],
      
                [0 0]     [1]
      [a](x0) = [0 0]x0 + [0]
     orientation:
      a#(b(b(x1))) = [1 1]x1 + [4] >= [0 0]x1 + [3] = a#(b(c(a(x1))))
      
      a#(b(b(x1))) = [1 1]x1 + [4] >= [0 0]x1 + [0] = a#(x1)
      
              [0 0]     [1]           
      a(x1) = [0 0]x1 + [0] >= x1 = x1
      
                    [1 1]     [4]    [1 1]     [4]                       
      a(b(b(x1))) = [1 1]x1 + [4] >= [1 1]x1 + [4] = b(b(a(b(c(a(x1))))))
      
                 [0 0]     [3]           
      b(c(x1)) = [0 0]x1 + [3] >= x1 = x1
     problem:
      DPs:
       
      TRS:
       a(x1) -> x1
       a(b(b(x1))) -> b(b(a(b(c(a(x1))))))
       b(c(x1)) -> x1
     Qed