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
  Usable Rule 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:
    P(p(x1)) -> x1
    r(p(x1)) -> p(p(r(P(x1))))
    r(r(x1)) -> x1
    r(P(P(x1))) -> P(P(r(x1)))
    r(R(x1)) -> x1
    p(P(x1)) -> x1
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     P(p(x1)) -> x1
    interpretation:
     [p#](x0) = 0,
     
     [P#](x0) = 0,
     
     [r#](x0) = x0 + 0,
     
     [R#](x0) = 1x0 + 12,
     
     [P](x0) = 3x0 + -2,
     
     [p](x0) = 13x0 + 2,
     
     [r](x0) = x0,
     
     [R](x0) = 4x0 + 10
    orientation:
     R#(x1) = 1x1 + 12 >= x1 + 0 = r#(x1)
     
     r#(p(x1)) = 13x1 + 2 >= 0 = P#(x1)
     
     r#(p(x1)) = 13x1 + 2 >= 3x1 + 0 = r#(P(x1))
     
     r#(p(x1)) = 13x1 + 2 >= 0 = p#(r(P(x1)))
     
     r#(p(x1)) = 13x1 + 2 >= 0 = p#(p(r(P(x1))))
     
     r#(P(P(x1))) = 6x1 + 1 >= x1 + 0 = r#(x1)
     
     r#(P(P(x1))) = 6x1 + 1 >= 0 = P#(r(x1))
     
     r#(P(P(x1))) = 6x1 + 1 >= 0 = P#(P(r(x1)))
     
     P(p(x1)) = 16x1 + 5 >= x1 = x1
     
     r(p(x1)) = 13x1 + 2 >= 29x1 + 24 = p(p(r(P(x1))))
     
     r(r(x1)) = x1 >= x1 = x1
     
     r(P(P(x1))) = 6x1 + 1 >= 6x1 + 1 = P(P(r(x1)))
     
     r(R(x1)) = 4x1 + 10 >= x1 = x1
     
     p(P(x1)) = 16x1 + 11 >= x1 = x1
    problem:
     DPs:
      
     TRS:
      P(p(x1)) -> x1
      r(p(x1)) -> p(p(r(P(x1))))
      r(r(x1)) -> x1
      r(P(P(x1))) -> P(P(r(x1)))
      r(R(x1)) -> x1
      p(P(x1)) -> x1
    Qed