YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   d#(c(x1)) -> a#(x1)
   d#(c(x1)) -> d#(a(x1))
   a#(c(x1)) -> a#(x1)
  TRS:
   a(x1) -> b(c(x1))
   a(b(x1)) -> b(a(x1))
   d(c(x1)) -> d(a(x1))
   a(c(x1)) -> c(a(x1))
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    d#(c(x1)) -> a#(x1)
    d#(c(x1)) -> d#(a(x1))
    a#(c(x1)) -> a#(x1)
   TRS:
    a(x1) -> b(c(x1))
    a(b(x1)) -> b(a(x1))
    d(c(x1)) -> d(a(x1))
    a(c(x1)) -> c(a(x1))
   graph:
    d#(c(x1)) -> d#(a(x1)) -> d#(c(x1)) -> d#(a(x1))
    d#(c(x1)) -> d#(a(x1)) -> d#(c(x1)) -> a#(x1)
    d#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1)
    d#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
    a#(b(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
    a#(c(x1)) -> a#(x1) -> a#(c(x1)) -> a#(x1)
    a#(c(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   CDG Processor:
    DPs:
     a#(b(x1)) -> a#(x1)
     d#(c(x1)) -> a#(x1)
     d#(c(x1)) -> d#(a(x1))
     a#(c(x1)) -> a#(x1)
    TRS:
     a(x1) -> b(c(x1))
     a(b(x1)) -> b(a(x1))
     d(c(x1)) -> d(a(x1))
     a(c(x1)) -> c(a(x1))
    graph:
     d#(c(x1)) -> d#(a(x1)) -> d#(c(x1)) -> a#(x1)
     d#(c(x1)) -> d#(a(x1)) -> d#(c(x1)) -> d#(a(x1))
    SCC Processor:
     #sccs: 1
     #rules: 1
     #arcs: 2/16
     DPs:
      d#(c(x1)) -> d#(a(x1))
     TRS:
      a(x1) -> b(c(x1))
      a(b(x1)) -> b(a(x1))
      d(c(x1)) -> d(a(x1))
      a(c(x1)) -> c(a(x1))
     KBO Processor:
      argument filtering:
       pi(a) = [0]
       pi(c) = [0]
       pi(b) = []
       pi(d) = 0
       pi(d#) = 0
      weight function:
       w0 = 1
       w(d) = w(b) = w(c) = 1
       w(d#) = w(a) = 0
      precedence:
       a > d# ~ d ~ b ~ c
      problem:
       DPs:
        
       TRS:
        a(x1) -> b(c(x1))
        a(b(x1)) -> b(a(x1))
        d(c(x1)) -> d(a(x1))
        a(c(x1)) -> c(a(x1))
      Qed