YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> b#(x1)
   a#(a(x1)) -> b#(b(x1))
   a#(a(x1)) -> b#(b(b(x1)))
   b#(b(x1)) -> c#(x1)
   b#(b(x1)) -> c#(c(x1))
   b#(b(x1)) -> c#(c(c(x1)))
   c#(d(d(x1))) -> a#(x1)
  TRS:
   a(a(x1)) -> b(b(b(x1)))
   b(b(x1)) -> c(c(c(x1)))
   c(c(x1)) -> d(d(d(x1)))
   b(x1) -> d(d(x1))
   c(d(d(x1))) -> a(x1)
  TDG Processor:
   DPs:
    a#(a(x1)) -> b#(x1)
    a#(a(x1)) -> b#(b(x1))
    a#(a(x1)) -> b#(b(b(x1)))
    b#(b(x1)) -> c#(x1)
    b#(b(x1)) -> c#(c(x1))
    b#(b(x1)) -> c#(c(c(x1)))
    c#(d(d(x1))) -> a#(x1)
   TRS:
    a(a(x1)) -> b(b(b(x1)))
    b(b(x1)) -> c(c(c(x1)))
    c(c(x1)) -> d(d(d(x1)))
    b(x1) -> d(d(x1))
    c(d(d(x1))) -> a(x1)
   graph:
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(b(x1)))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(x1))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(x1)
    b#(b(x1)) -> c#(c(c(x1))) -> c#(d(d(x1))) -> a#(x1)
    b#(b(x1)) -> c#(c(x1)) -> c#(d(d(x1))) -> a#(x1)
    b#(b(x1)) -> c#(x1) -> c#(d(d(x1))) -> a#(x1)
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(c(x1))
    a#(a(x1)) -> b#(b(b(x1))) -> b#(b(x1)) -> c#(x1)
    a#(a(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(c(x1))
    a#(a(x1)) -> b#(b(x1)) -> b#(b(x1)) -> c#(x1)
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(c(x1))
    a#(a(x1)) -> b#(x1) -> b#(b(x1)) -> c#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [c#](x0) = x0,
     
     [b#](x0) = 4x0,
     
     [a#](x0) = 8x0,
     
     [d](x0) = 4x0,
     
     [c](x0) = 6x0,
     
     [b](x0) = 9x0,
     
     [a](x0) = 14x0
    orientation:
     a#(a(x1)) = 22x1 >= 4x1 = b#(x1)
     
     a#(a(x1)) = 22x1 >= 13x1 = b#(b(x1))
     
     a#(a(x1)) = 22x1 >= 22x1 = b#(b(b(x1)))
     
     b#(b(x1)) = 13x1 >= x1 = c#(x1)
     
     b#(b(x1)) = 13x1 >= 6x1 = c#(c(x1))
     
     b#(b(x1)) = 13x1 >= 12x1 = c#(c(c(x1)))
     
     c#(d(d(x1))) = 8x1 >= 8x1 = a#(x1)
     
     a(a(x1)) = 28x1 >= 27x1 = b(b(b(x1)))
     
     b(b(x1)) = 18x1 >= 18x1 = c(c(c(x1)))
     
     c(c(x1)) = 12x1 >= 12x1 = d(d(d(x1)))
     
     b(x1) = 9x1 >= 8x1 = d(d(x1))
     
     c(d(d(x1))) = 14x1 >= 14x1 = a(x1)
    problem:
     DPs:
      a#(a(x1)) -> b#(b(b(x1)))
      c#(d(d(x1))) -> a#(x1)
     TRS:
      a(a(x1)) -> b(b(b(x1)))
      b(b(x1)) -> c(c(c(x1)))
      c(c(x1)) -> d(d(d(x1)))
      b(x1) -> d(d(x1))
      c(d(d(x1))) -> a(x1)
    EDG Processor:
     DPs:
      a#(a(x1)) -> b#(b(b(x1)))
      c#(d(d(x1))) -> a#(x1)
     TRS:
      a(a(x1)) -> b(b(b(x1)))
      b(b(x1)) -> c(c(c(x1)))
      c(c(x1)) -> d(d(d(x1)))
      b(x1) -> d(d(x1))
      c(d(d(x1))) -> a(x1)
     graph:
      c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(b(b(x1)))
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 1/4