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))
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [d#](x0) = x0 + 0,
       
       [d](x0) = 12x0 + 0,
       
       [b](x0) = -16x0 + 0,
       
       [c](x0) = 1x0 + 1,
       
       [a](x0) = x0 + 0
      orientation:
       d#(c(x1)) = 1x1 + 1 >= x1 + 0 = d#(a(x1))
       
       a(x1) = x1 + 0 >= -15x1 + 0 = b(c(x1))
       
       a(b(x1)) = -16x1 + 0 >= -16x1 + 0 = b(a(x1))
       
       d(c(x1)) = 13x1 + 13 >= 12x1 + 12 = d(a(x1))
       
       a(c(x1)) = 1x1 + 1 >= 1x1 + 1 = c(a(x1))
      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