YES

Problem:
 R(x1) -> r(x1)
 r(p(x1)) -> p(p(r(P(x1))))
 r(r(x1)) -> x1
 r(P(P(x1))) -> P(P(r(x1)))
 p(P(x1)) -> x1
 P(p(x1)) -> x1
 r(R(x1)) -> x1
 R(r(x1)) -> x1

Proof:
 DP Processor:
  DPs:
   R#(x1) -> r#(x1)
   r#(p(x1)) -> P#(x1)
   r#(p(x1)) -> r#(P(x1))
   r#(p(x1)) -> p#(r(P(x1)))
   r#(p(x1)) -> p#(p(r(P(x1))))
   r#(P(P(x1))) -> r#(x1)
   r#(P(P(x1))) -> P#(r(x1))
   r#(P(P(x1))) -> P#(P(r(x1)))
  TRS:
   R(x1) -> r(x1)
   r(p(x1)) -> p(p(r(P(x1))))
   r(r(x1)) -> x1
   r(P(P(x1))) -> P(P(r(x1)))
   p(P(x1)) -> x1
   P(p(x1)) -> x1
   r(R(x1)) -> x1
   R(r(x1)) -> x1
  TDG Processor:
   DPs:
    R#(x1) -> r#(x1)
    r#(p(x1)) -> P#(x1)
    r#(p(x1)) -> r#(P(x1))
    r#(p(x1)) -> p#(r(P(x1)))
    r#(p(x1)) -> p#(p(r(P(x1))))
    r#(P(P(x1))) -> r#(x1)
    r#(P(P(x1))) -> P#(r(x1))
    r#(P(P(x1))) -> P#(P(r(x1)))
   TRS:
    R(x1) -> r(x1)
    r(p(x1)) -> p(p(r(P(x1))))
    r(r(x1)) -> x1
    r(P(P(x1))) -> P(P(r(x1)))
    p(P(x1)) -> x1
    P(p(x1)) -> x1
    r(R(x1)) -> x1
    R(r(x1)) -> x1
   graph:
    r#(P(P(x1))) -> r#(x1) -> r#(P(P(x1))) -> P#(P(r(x1)))
    r#(P(P(x1))) -> r#(x1) -> r#(P(P(x1))) -> P#(r(x1))
    r#(P(P(x1))) -> r#(x1) -> r#(P(P(x1))) -> r#(x1)
    r#(P(P(x1))) -> r#(x1) -> r#(p(x1)) -> p#(p(r(P(x1))))
    r#(P(P(x1))) -> r#(x1) -> r#(p(x1)) -> p#(r(P(x1)))
    r#(P(P(x1))) -> r#(x1) -> r#(p(x1)) -> r#(P(x1))
    r#(P(P(x1))) -> r#(x1) -> r#(p(x1)) -> P#(x1)
    r#(p(x1)) -> r#(P(x1)) -> r#(P(P(x1))) -> P#(P(r(x1)))
    r#(p(x1)) -> r#(P(x1)) -> r#(P(P(x1))) -> P#(r(x1))
    r#(p(x1)) -> r#(P(x1)) -> r#(P(P(x1))) -> r#(x1)
    r#(p(x1)) -> r#(P(x1)) -> r#(p(x1)) -> p#(p(r(P(x1))))
    r#(p(x1)) -> r#(P(x1)) -> r#(p(x1)) -> p#(r(P(x1)))
    r#(p(x1)) -> r#(P(x1)) -> r#(p(x1)) -> r#(P(x1))
    r#(p(x1)) -> r#(P(x1)) -> r#(p(x1)) -> P#(x1)
    R#(x1) -> r#(x1) -> r#(P(P(x1))) -> P#(P(r(x1)))
    R#(x1) -> r#(x1) -> r#(P(P(x1))) -> P#(r(x1))
    R#(x1) -> r#(x1) -> r#(P(P(x1))) -> r#(x1)
    R#(x1) -> r#(x1) -> r#(p(x1)) -> p#(p(r(P(x1))))
    R#(x1) -> r#(x1) -> r#(p(x1)) -> p#(r(P(x1)))
    R#(x1) -> r#(x1) -> r#(p(x1)) -> r#(P(x1))
    R#(x1) -> r#(x1) -> r#(p(x1)) -> P#(x1)
   CDG Processor:
    DPs:
     R#(x1) -> r#(x1)
     r#(p(x1)) -> P#(x1)
     r#(p(x1)) -> r#(P(x1))
     r#(p(x1)) -> p#(r(P(x1)))
     r#(p(x1)) -> p#(p(r(P(x1))))
     r#(P(P(x1))) -> r#(x1)
     r#(P(P(x1))) -> P#(r(x1))
     r#(P(P(x1))) -> P#(P(r(x1)))
    TRS:
     R(x1) -> r(x1)
     r(p(x1)) -> p(p(r(P(x1))))
     r(r(x1)) -> x1
     r(P(P(x1))) -> P(P(r(x1)))
     p(P(x1)) -> x1
     P(p(x1)) -> x1
     r(R(x1)) -> x1
     R(r(x1)) -> x1
    graph:
     
    Qed