YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> b#(c(x1))
   b#(b(x1)) -> d#(x1)
   b#(b(x1)) -> c#(d(x1))
   c#(c(x1)) -> d#(x1)
   c#(c(x1)) -> d#(d(x1))
   c#(c(x1)) -> d#(d(d(x1)))
   d#(d(d(x1))) -> c#(x1)
   d#(d(d(x1))) -> a#(c(x1))
  TRS:
   a(a(x1)) -> b(c(x1))
   b(b(x1)) -> c(d(x1))
   c(c(x1)) -> d(d(d(x1)))
   d(d(d(x1))) -> a(c(x1))
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> b#(c(x1))
    b#(b(x1)) -> d#(x1)
    b#(b(x1)) -> c#(d(x1))
    c#(c(x1)) -> d#(x1)
    c#(c(x1)) -> d#(d(x1))
    c#(c(x1)) -> d#(d(d(x1)))
    d#(d(d(x1))) -> c#(x1)
    d#(d(d(x1))) -> a#(c(x1))
   TRS:
    a(a(x1)) -> b(c(x1))
    b(b(x1)) -> c(d(x1))
    c(c(x1)) -> d(d(d(x1)))
    d(d(d(x1))) -> a(c(x1))
   graph:
    d#(d(d(x1))) -> c#(x1) -> c#(c(x1)) -> d#(d(d(x1)))
    d#(d(d(x1))) -> c#(x1) -> c#(c(x1)) -> d#(d(x1))
    d#(d(d(x1))) -> c#(x1) -> c#(c(x1)) -> d#(x1)
    d#(d(d(x1))) -> a#(c(x1)) -> a#(a(x1)) -> b#(c(x1))
    d#(d(d(x1))) -> a#(c(x1)) -> a#(a(x1)) -> c#(x1)
    b#(b(x1)) -> d#(x1) -> d#(d(d(x1))) -> a#(c(x1))
    b#(b(x1)) -> d#(x1) -> d#(d(d(x1))) -> c#(x1)
    b#(b(x1)) -> c#(d(x1)) -> c#(c(x1)) -> d#(d(d(x1)))
    b#(b(x1)) -> c#(d(x1)) -> c#(c(x1)) -> d#(d(x1))
    b#(b(x1)) -> c#(d(x1)) -> c#(c(x1)) -> d#(x1)
    c#(c(x1)) -> d#(d(d(x1))) -> d#(d(d(x1))) -> a#(c(x1))
    c#(c(x1)) -> d#(d(d(x1))) -> d#(d(d(x1))) -> c#(x1)
    c#(c(x1)) -> d#(d(x1)) -> d#(d(d(x1))) -> a#(c(x1))
    c#(c(x1)) -> d#(d(x1)) -> d#(d(d(x1))) -> c#(x1)
    c#(c(x1)) -> d#(x1) -> d#(d(d(x1))) -> a#(c(x1))
    c#(c(x1)) -> d#(x1) -> d#(d(d(x1))) -> c#(x1)
    a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> c#(d(x1))
    a#(a(x1)) -> b#(c(x1)) -> b#(b(x1)) -> d#(x1)
    a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> d#(d(d(x1)))
    a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> d#(d(x1))
    a#(a(x1)) -> c#(x1) -> c#(c(x1)) -> d#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [d#](x0) = 5x0,
     
     [b#](x0) = 6x0 + 12,
     
     [c#](x0) = 8x0 + 0,
     
     [a#](x0) = 8x0 + 0,
     
     [d](x0) = 6x0 + 1,
     
     [b](x0) = 8x0 + 8,
     
     [c](x0) = 9x0 + 4,
     
     [a](x0) = 9x0 + 6
    orientation:
     a#(a(x1)) = 17x1 + 14 >= 8x1 + 0 = c#(x1)
     
     a#(a(x1)) = 17x1 + 14 >= 15x1 + 12 = b#(c(x1))
     
     b#(b(x1)) = 14x1 + 14 >= 5x1 = d#(x1)
     
     b#(b(x1)) = 14x1 + 14 >= 14x1 + 9 = c#(d(x1))
     
     c#(c(x1)) = 17x1 + 12 >= 5x1 = d#(x1)
     
     c#(c(x1)) = 17x1 + 12 >= 11x1 + 6 = d#(d(x1))
     
     c#(c(x1)) = 17x1 + 12 >= 17x1 + 12 = d#(d(d(x1)))
     
     d#(d(d(x1))) = 17x1 + 12 >= 8x1 + 0 = c#(x1)
     
     d#(d(d(x1))) = 17x1 + 12 >= 17x1 + 12 = a#(c(x1))
     
     a(a(x1)) = 18x1 + 15 >= 17x1 + 12 = b(c(x1))
     
     b(b(x1)) = 16x1 + 16 >= 15x1 + 10 = c(d(x1))
     
     c(c(x1)) = 18x1 + 13 >= 18x1 + 13 = d(d(d(x1)))
     
     d(d(d(x1))) = 18x1 + 13 >= 18x1 + 13 = a(c(x1))
    problem:
     DPs:
      b#(b(x1)) -> c#(d(x1))
      c#(c(x1)) -> d#(d(d(x1)))
      d#(d(d(x1))) -> a#(c(x1))
     TRS:
      a(a(x1)) -> b(c(x1))
      b(b(x1)) -> c(d(x1))
      c(c(x1)) -> d(d(d(x1)))
      d(d(d(x1))) -> a(c(x1))
    EDG Processor:
     DPs:
      b#(b(x1)) -> c#(d(x1))
      c#(c(x1)) -> d#(d(d(x1)))
      d#(d(d(x1))) -> a#(c(x1))
     TRS:
      a(a(x1)) -> b(c(x1))
      b(b(x1)) -> c(d(x1))
      c(c(x1)) -> d(d(d(x1)))
      d(d(d(x1))) -> a(c(x1))
     graph:
      b#(b(x1)) -> c#(d(x1)) -> c#(c(x1)) -> d#(d(d(x1)))
      c#(c(x1)) -> d#(d(d(x1))) -> d#(d(d(x1))) -> a#(c(x1))
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/9