YES
Input TRS:
      1: R(x1) -> r(x1)
      2: r(p(x1)) -> p(p(r(P(x1))))
      3: r(r(x1)) -> x1
      4: r(P(P(x1))) -> P(P(r(x1)))
      5: p(P(x1)) -> x1
      6: P(p(x1)) -> x1
      7: r(R(x1)) -> x1
      8: R(r(x1)) -> x1
Number of strict rules: 8
Direct POLO(bPol) ... removes: 8 1 3 7
      r 	w: x1 + 1
      p 	w: x1
      R 	w: x1 + 2
      P 	w: x1
Number of strict rules: 4
Direct POLO(bPol) ... removes: 5 6
      r 	w: 2 * x1 + 30613
      p 	w: x1 + 33954
      R 	w: 2 * x1 + 30614
      P 	w: x1
Number of strict rules: 2
Direct POLO(bPol) ... failed.
Uncurrying r
      2: r^1_p(x1) -> p(p(r^1_P(x1)))
      4: r^1_P(P(x1)) -> P(P(r(x1)))
      9: r(p(_1)) ->= r^1_p(_1)
      10: r(P(_1)) ->= r^1_P(_1)
Number of strict rules: 2
Direct POLO(bPol) ... removes: 10
      r^1_p	w: 2 * x1 + 41077
      r 	w: 2 * x1 + 1
      p 	w: x1 + 20538
      R 	w: x1
      P 	w: x1 + 1
      r^1_P	w: 2 * x1 + 1
Number of strict rules: 2
Direct POLO(bPol) ... failed.
Dependency Pairs:
     #1: #r^1_p(x1) -> #r^1_P(x1)
     #2: #r(p(_1)) ->? #r^1_p(_1)
     #3: #r^1_P(P(x1)) -> #r(x1)
Number of SCCs: 1, DPs: 3
  SCC { #1..3 }
POLO(Sum)... succeeded.
      r^1_p	w: 0
      r 	w: 0
      #r^1_p	w: x1 + 1
      p 	w: x1 + 2
      #r^1_P	w: x1
      R 	w: 0
      #r 	w: x1
      P 	w: x1 + 1
      r^1_P	w: 0
    USABLE RULES: { }
    Removed DPs: #1..3
Number of SCCs: 0, DPs: 0