YES

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

Proof:
 DP Processor:
  DPs:
   a#(b(x1)) -> r#(x1)
   r#(a(x1)) -> r#(x1)
   r#(a(x1)) -> d#(r(x1))
   r#(x1) -> d#(x1)
   d#(a(x1)) -> d#(x1)
   d#(a(x1)) -> a#(d(x1))
   d#(a(x1)) -> a#(a(d(x1)))
   d#(x1) -> a#(x1)
  TRS:
   a(b(x1)) -> b(r(x1))
   r(a(x1)) -> d(r(x1))
   r(x1) -> d(x1)
   d(a(x1)) -> a(a(d(x1)))
   d(x1) -> a(x1)
  TDG Processor:
   DPs:
    a#(b(x1)) -> r#(x1)
    r#(a(x1)) -> r#(x1)
    r#(a(x1)) -> d#(r(x1))
    r#(x1) -> d#(x1)
    d#(a(x1)) -> d#(x1)
    d#(a(x1)) -> a#(d(x1))
    d#(a(x1)) -> a#(a(d(x1)))
    d#(x1) -> a#(x1)
   TRS:
    a(b(x1)) -> b(r(x1))
    r(a(x1)) -> d(r(x1))
    r(x1) -> d(x1)
    d(a(x1)) -> a(a(d(x1)))
    d(x1) -> a(x1)
   graph:
    d#(a(x1)) -> d#(x1) -> d#(x1) -> a#(x1)
    d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> a#(a(d(x1)))
    d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> a#(d(x1))
    d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> d#(x1)
    d#(a(x1)) -> a#(d(x1)) -> a#(b(x1)) -> r#(x1)
    d#(a(x1)) -> a#(a(d(x1))) -> a#(b(x1)) -> r#(x1)
    d#(x1) -> a#(x1) -> a#(b(x1)) -> r#(x1)
    r#(a(x1)) -> d#(r(x1)) -> d#(x1) -> a#(x1)
    r#(a(x1)) -> d#(r(x1)) -> d#(a(x1)) -> a#(a(d(x1)))
    r#(a(x1)) -> d#(r(x1)) -> d#(a(x1)) -> a#(d(x1))
    r#(a(x1)) -> d#(r(x1)) -> d#(a(x1)) -> d#(x1)
    r#(a(x1)) -> r#(x1) -> r#(x1) -> d#(x1)
    r#(a(x1)) -> r#(x1) -> r#(a(x1)) -> d#(r(x1))
    r#(a(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
    r#(x1) -> d#(x1) -> d#(x1) -> a#(x1)
    r#(x1) -> d#(x1) -> d#(a(x1)) -> a#(a(d(x1)))
    r#(x1) -> d#(x1) -> d#(a(x1)) -> a#(d(x1))
    r#(x1) -> d#(x1) -> d#(a(x1)) -> d#(x1)
    a#(b(x1)) -> r#(x1) -> r#(x1) -> d#(x1)
    a#(b(x1)) -> r#(x1) -> r#(a(x1)) -> d#(r(x1))
    a#(b(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
   Arctic Interpretation Processor:
    dimension: 1
    interpretation:
     [d#](x0) = x0,
     
     [r#](x0) = 8x0,
     
     [a#](x0) = x0,
     
     [d](x0) = x0,
     
     [r](x0) = x0,
     
     [a](x0) = x0,
     
     [b](x0) = 8x0
    orientation:
     a#(b(x1)) = 8x1 >= 8x1 = r#(x1)
     
     r#(a(x1)) = 8x1 >= 8x1 = r#(x1)
     
     r#(a(x1)) = 8x1 >= x1 = d#(r(x1))
     
     r#(x1) = 8x1 >= x1 = d#(x1)
     
     d#(a(x1)) = x1 >= x1 = d#(x1)
     
     d#(a(x1)) = x1 >= x1 = a#(d(x1))
     
     d#(a(x1)) = x1 >= x1 = a#(a(d(x1)))
     
     d#(x1) = x1 >= x1 = a#(x1)
     
     a(b(x1)) = 8x1 >= 8x1 = b(r(x1))
     
     r(a(x1)) = x1 >= x1 = d(r(x1))
     
     r(x1) = x1 >= x1 = d(x1)
     
     d(a(x1)) = x1 >= x1 = a(a(d(x1)))
     
     d(x1) = x1 >= x1 = a(x1)
    problem:
     DPs:
      a#(b(x1)) -> r#(x1)
      r#(a(x1)) -> r#(x1)
      d#(a(x1)) -> d#(x1)
      d#(a(x1)) -> a#(d(x1))
      d#(a(x1)) -> a#(a(d(x1)))
      d#(x1) -> a#(x1)
     TRS:
      a(b(x1)) -> b(r(x1))
      r(a(x1)) -> d(r(x1))
      r(x1) -> d(x1)
      d(a(x1)) -> a(a(d(x1)))
      d(x1) -> a(x1)
    EDG Processor:
     DPs:
      a#(b(x1)) -> r#(x1)
      r#(a(x1)) -> r#(x1)
      d#(a(x1)) -> d#(x1)
      d#(a(x1)) -> a#(d(x1))
      d#(a(x1)) -> a#(a(d(x1)))
      d#(x1) -> a#(x1)
     TRS:
      a(b(x1)) -> b(r(x1))
      r(a(x1)) -> d(r(x1))
      r(x1) -> d(x1)
      d(a(x1)) -> a(a(d(x1)))
      d(x1) -> a(x1)
     graph:
      d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> d#(x1)
      d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> a#(d(x1))
      d#(a(x1)) -> d#(x1) -> d#(a(x1)) -> a#(a(d(x1)))
      d#(a(x1)) -> d#(x1) -> d#(x1) -> a#(x1)
      d#(a(x1)) -> a#(d(x1)) -> a#(b(x1)) -> r#(x1)
      d#(a(x1)) -> a#(a(d(x1))) -> a#(b(x1)) -> r#(x1)
      d#(x1) -> a#(x1) -> a#(b(x1)) -> r#(x1)
      r#(a(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
      a#(b(x1)) -> r#(x1) -> r#(a(x1)) -> r#(x1)
     CDG Processor:
      DPs:
       a#(b(x1)) -> r#(x1)
       r#(a(x1)) -> r#(x1)
       d#(a(x1)) -> d#(x1)
       d#(a(x1)) -> a#(d(x1))
       d#(a(x1)) -> a#(a(d(x1)))
       d#(x1) -> a#(x1)
      TRS:
       a(b(x1)) -> b(r(x1))
       r(a(x1)) -> d(r(x1))
       r(x1) -> d(x1)
       d(a(x1)) -> a(a(d(x1)))
       d(x1) -> a(x1)
      graph:
       d#(a(x1)) -> d#(x1) -> d#(x1) -> a#(x1)
       d#(a(x1)) -> a#(d(x1)) -> a#(b(x1)) -> r#(x1)
       d#(a(x1)) -> a#(a(d(x1))) -> a#(b(x1)) -> r#(x1)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 3/36