YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> a#(x1)
   c#(b(x1)) -> c#(x1)
   c#(b(x1)) -> c#(c(x1))
  TRS:
   a(b(x1)) -> b(b(a(x1)))
   c(b(x1)) -> b(c(c(x1)))
  TDG Processor:
   DPs:
    a#(b(x1)) -> a#(x1)
    c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> c#(c(x1))
   TRS:
    a(b(x1)) -> b(b(a(x1)))
    c(b(x1)) -> b(c(c(x1)))
   graph:
    c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(c(x1))
    c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(x1)
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(c(x1))
    c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
    a#(b(x1)) -> a#(x1) -> a#(b(x1)) -> a#(x1)
   CDG Processor:
    DPs:
     a#(b(x1)) -> a#(x1)
     c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> c#(c(x1))
    TRS:
     a(b(x1)) -> b(b(a(x1)))
     c(b(x1)) -> b(c(c(x1)))
    graph:
     c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> c#(c(x1)) -> c#(b(x1)) -> c#(c(x1))
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(x1)
     c#(b(x1)) -> c#(x1) -> c#(b(x1)) -> c#(c(x1))
    SCC Processor:
     #sccs: 1
     #rules: 2
     #arcs: 4/9
     DPs:
      c#(b(x1)) -> c#(c(x1))
      c#(b(x1)) -> c#(x1)
     TRS:
      a(b(x1)) -> b(b(a(x1)))
      c(b(x1)) -> b(c(c(x1)))
     LPO Processor:
      argument filtering:
       pi(b) = [0]
       pi(a) = [0]
       pi(c) = 0
       pi(c#) = 0
      precedence:
       a > c# ~ c ~ b
      problem:
       DPs:
        
       TRS:
        a(b(x1)) -> b(b(a(x1)))
        c(b(x1)) -> b(c(c(x1)))
      Qed