YES

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

Proof:
 DP Processor:
  DPs:
   a#(a(x1)) -> c#(x1)
   a#(a(x1)) -> c#(c(x1))
   a#(a(x1)) -> c#(c(c(x1)))
   a#(a(x1)) -> b#(c(c(c(x1))))
   a#(x1) -> c#(d(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(c(c(c(x1))))
   b(c(x1)) -> d(d(d(d(x1))))
   a(x1) -> d(c(d(x1)))
   b(b(x1)) -> c(c(c(x1)))
   c(c(x1)) -> d(d(d(x1)))
   c(d(d(x1))) -> a(x1)
  TDG Processor:
   DPs:
    a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> c#(c(x1))
    a#(a(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(c(c(c(x1))))
    a#(x1) -> c#(d(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(c(c(c(x1))))
    b(c(x1)) -> d(d(d(d(x1))))
    a(x1) -> d(c(d(x1)))
    b(b(x1)) -> c(c(c(x1)))
    c(c(x1)) -> d(d(d(x1)))
    c(d(d(x1))) -> a(x1)
   graph:
    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)
    c#(d(d(x1))) -> a#(x1) -> a#(x1) -> c#(d(x1))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> b#(c(c(c(x1))))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> c#(c(c(x1)))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> c#(c(x1))
    c#(d(d(x1))) -> a#(x1) -> a#(a(x1)) -> c#(x1)
    a#(a(x1)) -> b#(c(c(c(x1)))) -> b#(b(x1)) -> c#(c(c(x1)))
    a#(a(x1)) -> b#(c(c(c(x1)))) -> b#(b(x1)) -> c#(c(x1))
    a#(a(x1)) -> b#(c(c(c(x1)))) -> b#(b(x1)) -> c#(x1)
    a#(a(x1)) -> c#(c(c(x1))) -> c#(d(d(x1))) -> a#(x1)
    a#(a(x1)) -> c#(c(x1)) -> c#(d(d(x1))) -> a#(x1)
    a#(a(x1)) -> c#(x1) -> c#(d(d(x1))) -> a#(x1)
    a#(x1) -> c#(d(x1)) -> c#(d(d(x1))) -> a#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [b#](x0) = 1x0,
     
     [c#](x0) = x0,
     
     [a#](x0) = 3x0,
     
     [d](x0) = 2x0,
     
     [b](x0) = 5x0,
     
     [c](x0) = 3x0,
     
     [a](x0) = 7x0
    orientation:
     a#(a(x1)) = 10x1 >= x1 = c#(x1)
     
     a#(a(x1)) = 10x1 >= 3x1 = c#(c(x1))
     
     a#(a(x1)) = 10x1 >= 6x1 = c#(c(c(x1)))
     
     a#(a(x1)) = 10x1 >= 10x1 = b#(c(c(c(x1))))
     
     a#(x1) = 3x1 >= 2x1 = c#(d(x1))
     
     b#(b(x1)) = 6x1 >= x1 = c#(x1)
     
     b#(b(x1)) = 6x1 >= 3x1 = c#(c(x1))
     
     b#(b(x1)) = 6x1 >= 6x1 = c#(c(c(x1)))
     
     c#(d(d(x1))) = 4x1 >= 3x1 = a#(x1)
     
     a(a(x1)) = 14x1 >= 14x1 = b(c(c(c(x1))))
     
     b(c(x1)) = 8x1 >= 8x1 = d(d(d(d(x1))))
     
     a(x1) = 7x1 >= 7x1 = d(c(d(x1)))
     
     b(b(x1)) = 10x1 >= 9x1 = c(c(c(x1)))
     
     c(c(x1)) = 6x1 >= 6x1 = d(d(d(x1)))
     
     c(d(d(x1))) = 7x1 >= 7x1 = a(x1)
    problem:
     DPs:
      a#(a(x1)) -> b#(c(c(c(x1))))
      b#(b(x1)) -> c#(c(c(x1)))
     TRS:
      a(a(x1)) -> b(c(c(c(x1))))
      b(c(x1)) -> d(d(d(d(x1))))
      a(x1) -> d(c(d(x1)))
      b(b(x1)) -> c(c(c(x1)))
      c(c(x1)) -> d(d(d(x1)))
      c(d(d(x1))) -> a(x1)
    EDG Processor:
     DPs:
      a#(a(x1)) -> b#(c(c(c(x1))))
      b#(b(x1)) -> c#(c(c(x1)))
     TRS:
      a(a(x1)) -> b(c(c(c(x1))))
      b(c(x1)) -> d(d(d(d(x1))))
      a(x1) -> d(c(d(x1)))
      b(b(x1)) -> c(c(c(x1)))
      c(c(x1)) -> d(d(d(x1)))
      c(d(d(x1))) -> a(x1)
     graph:
      a#(a(x1)) -> b#(c(c(c(x1)))) -> b#(b(x1)) -> c#(c(c(x1)))
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 1/4