YES

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

Proof:
 WPO Processor:
  algebra: Sum
  weight function:
   w0 = 0
   w(b) = w(c) = w(a) = 3
   w(d) = 2
  status function:
   st(d) = st(b) = st(c) = st(a) = [0]
  precedence:
   c > d > a > b
  problem:
   
  Qed