YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(a(x1))) -> b#(x1)
   a#(a(a(x1))) -> b#(b(x1))
   b#(b(b(x1))) -> c#(x1)
   c#(x1) -> d#(x1)
   c#(x1) -> d#(d(x1))
   d#(x1) -> a#(x1)
   d#(x1) -> a#(a(x1))
  TRS:
   a(a(a(x1))) -> b(b(x1))
   b(b(b(x1))) -> c(x1)
   c(x1) -> d(d(x1))
   d(x1) -> a(a(x1))
  TDG Processor:
   DPs:
    a#(a(a(x1))) -> b#(x1)
    a#(a(a(x1))) -> b#(b(x1))
    b#(b(b(x1))) -> c#(x1)
    c#(x1) -> d#(x1)
    c#(x1) -> d#(d(x1))
    d#(x1) -> a#(x1)
    d#(x1) -> a#(a(x1))
   TRS:
    a(a(a(x1))) -> b(b(x1))
    b(b(b(x1))) -> c(x1)
    c(x1) -> d(d(x1))
    d(x1) -> a(a(x1))
   graph:
    d#(x1) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1))
    d#(x1) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(x1)
    d#(x1) -> a#(x1) -> a#(a(a(x1))) -> b#(b(x1))
    d#(x1) -> a#(x1) -> a#(a(a(x1))) -> b#(x1)
    c#(x1) -> d#(d(x1)) -> d#(x1) -> a#(a(x1))
    c#(x1) -> d#(d(x1)) -> d#(x1) -> a#(x1)
    c#(x1) -> d#(x1) -> d#(x1) -> a#(a(x1))
    c#(x1) -> d#(x1) -> d#(x1) -> a#(x1)
    b#(b(b(x1))) -> c#(x1) -> c#(x1) -> d#(d(x1))
    b#(b(b(x1))) -> c#(x1) -> c#(x1) -> d#(x1)
    a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> c#(x1)
    a#(a(a(x1))) -> b#(x1) -> b#(b(b(x1))) -> c#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [d#](x0) = 2x0,
     
     [c#](x0) = 7x0,
     
     [b#](x0) = 1x0,
     
     [a#](x0) = x0,
     
     [d](x0) = 4x0,
     
     [c](x0) = 8x0,
     
     [b](x0) = 3x0,
     
     [a](x0) = 2x0
    orientation:
     a#(a(a(x1))) = 4x1 >= 1x1 = b#(x1)
     
     a#(a(a(x1))) = 4x1 >= 4x1 = b#(b(x1))
     
     b#(b(b(x1))) = 7x1 >= 7x1 = c#(x1)
     
     c#(x1) = 7x1 >= 2x1 = d#(x1)
     
     c#(x1) = 7x1 >= 6x1 = d#(d(x1))
     
     d#(x1) = 2x1 >= x1 = a#(x1)
     
     d#(x1) = 2x1 >= 2x1 = a#(a(x1))
     
     a(a(a(x1))) = 6x1 >= 6x1 = b(b(x1))
     
     b(b(b(x1))) = 9x1 >= 8x1 = c(x1)
     
     c(x1) = 8x1 >= 8x1 = d(d(x1))
     
     d(x1) = 4x1 >= 4x1 = a(a(x1))
    problem:
     DPs:
      a#(a(a(x1))) -> b#(b(x1))
      b#(b(b(x1))) -> c#(x1)
      d#(x1) -> a#(a(x1))
     TRS:
      a(a(a(x1))) -> b(b(x1))
      b(b(b(x1))) -> c(x1)
      c(x1) -> d(d(x1))
      d(x1) -> a(a(x1))
    EDG Processor:
     DPs:
      a#(a(a(x1))) -> b#(b(x1))
      b#(b(b(x1))) -> c#(x1)
      d#(x1) -> a#(a(x1))
     TRS:
      a(a(a(x1))) -> b(b(x1))
      b(b(b(x1))) -> c(x1)
      c(x1) -> d(d(x1))
      d(x1) -> a(a(x1))
     graph:
      d#(x1) -> a#(a(x1)) -> a#(a(a(x1))) -> b#(b(x1))
      a#(a(a(x1))) -> b#(b(x1)) -> b#(b(b(x1))) -> c#(x1)
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/9