NO

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   a#(b(x1)) -> a#(a(x1))
   a#(b(x1)) -> c#(a(a(x1)))
  TRS:
   a(x1) -> x1
   a(b(x1)) -> x1
   a(b(x1)) -> b(c(a(a(x1))))
   c(c(x1)) -> b(x1)
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> c#(a(a(x1)))
   TRS:
    a(x1) -> x1
    a(b(x1)) -> x1
    a(b(x1)) -> b(c(a(a(x1))))
    c(c(x1)) -> b(x1)
   graph:
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> c#(a(a(x1)))
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(a(x1)) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> c#(a(a(x1)))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(a(x1))
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   SCC Processor:
    #sccs: 1
    #rules: 2
    #arcs: 6/9
    DPs:
     a#(b(x1)) -> a#(a(x1))
     a#(b(x1)) -> a#(x1)
    TRS:
     a(x1) -> x1
     a(b(x1)) -> x1
     a(b(x1)) -> b(c(a(a(x1))))
     c(c(x1)) -> b(x1)
    Loop Processor:
     loop length: 11
     terms:
      a(b(c(c(a(a(b(a(x1))))))))
      a(b(c(c(a(b(c(a(a(a(x1))))))))))
      a(b(c(c(a(b(c(a(a(x1)))))))))
      b(c(a(a(c(c(a(b(c(a(a(x1)))))))))))
      b(c(a(a(c(c(b(c(a(a(c(a(a(x1)))))))))))))
      b(c(a(a(b(b(c(a(a(c(a(a(x1))))))))))))
      b(c(a(a(b(b(c(a(c(a(a(x1)))))))))))
      b(c(a(a(b(b(c(c(a(a(x1))))))))))
      b(c(a(a(b(b(b(a(a(x1)))))))))
      b(c(a(b(c(a(a(b(b(a(a(x1)))))))))))
      b(c(a(b(c(a(b(c(a(a(b(a(a(x1)))))))))))))
     context: b(c([]))
     substitution:
      x1 -> a(x1)
     Qed