YES
Input TRS:
      1: p(0(x1)) -> 0(s(s(p(x1))))
      2: p(s(x1)) -> x1
      3: p(p(s(x1))) -> p(x1)
      4: f(s(x1)) -> p(s(g(p(s(s(x1))))))
      5: g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1)))))))))))
      6: j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1)))))))))))
      7: half(0(x1)) -> 0(s(s(half(p(s(p(s(x1))))))))
      8: half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
      9: rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))
Number of strict rules: 9
Direct POLO(bPol) ... failed.
Uncurrying j p
      1: p^1_0(x1) -> 0(s(s(p(x1))))
      2: p^1_s(x1) -> x1
      3: p(p^1_s(x1)) -> p(x1)
      4: f(s(x1)) -> p^1_s(g(p^1_s(s(x1))))
      5: g(s(x1)) -> p(p^1_s(s(s(j^1_s(p^1_s(p^1_s(x1)))))))
      6: j^1_s(x1) -> p^1_s(s(p^1_s(f(p^1_s(p(p^1_s(x1)))))))
      7: half(0(x1)) -> 0(s(s(half(p^1_s(p^1_s(x1))))))
      8: half(s(s(x1))) -> s(half(p(p^1_s(s(x1)))))
      9: rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))
      10: p(0(_1)) ->= p^1_0(_1)
      11: p(s(_1)) ->= p^1_s(_1)
      12: j(s(_1)) ->= j^1_s(_1)
Number of strict rules: 9
Direct POLO(bPol) ... removes: 12
      s 	w: x1
      p^1_0	w: x1
      rd	w: x1 + 1
      p^1_s	w: x1
      f 	w: x1 + 1
      half	w: x1 + 1
      p 	w: x1
      0 	w: x1
      j^1_s	w: x1 + 1
      j 	w: x1 + 118
      g 	w: x1 + 1
Number of strict rules: 9
Direct POLO(bPol) ... failed.
Dependency Pairs:
     #1: #j^1_s(x1) -> #p^1_s(s(p^1_s(f(p^1_s(p(p^1_s(x1)))))))
     #2: #j^1_s(x1) -> #p^1_s(f(p^1_s(p(p^1_s(x1)))))
     #3: #j^1_s(x1) -> #f(p^1_s(p(p^1_s(x1))))
     #4: #j^1_s(x1) -> #p^1_s(p(p^1_s(x1)))
     #5: #j^1_s(x1) -> #p(p^1_s(x1))
     #6: #j^1_s(x1) -> #p^1_s(x1)
     #7: #rd(0(x1)) -> #rd(x1)
     #8: #p(s(_1)) ->? #p^1_s(_1)
     #9: #half(0(x1)) -> #half(p^1_s(p^1_s(x1)))
     #10: #half(0(x1)) -> #p^1_s(p^1_s(x1))
     #11: #half(0(x1)) -> #p^1_s(x1)
     #12: #p(0(_1)) ->? #p^1_0(_1)
     #13: #g(s(x1)) -> #p(p^1_s(s(s(j^1_s(p^1_s(p^1_s(x1)))))))
     #14: #g(s(x1)) -> #p^1_s(s(s(j^1_s(p^1_s(p^1_s(x1))))))
     #15: #g(s(x1)) -> #j^1_s(p^1_s(p^1_s(x1)))
     #16: #g(s(x1)) -> #p^1_s(p^1_s(x1))
     #17: #g(s(x1)) -> #p^1_s(x1)
     #18: #p(p^1_s(x1)) -> #p(x1)
     #19: #p^1_0(x1) -> #p(x1)
     #20: #half(s(s(x1))) -> #half(p(p^1_s(s(x1))))
     #21: #half(s(s(x1))) -> #p(p^1_s(s(x1)))
     #22: #half(s(s(x1))) -> #p^1_s(s(x1))
     #23: #f(s(x1)) -> #p^1_s(g(p^1_s(s(x1))))
     #24: #f(s(x1)) -> #g(p^1_s(s(x1)))
     #25: #f(s(x1)) -> #p^1_s(s(x1))
Number of SCCs: 4, DPs: 9
  SCC { #7 }
POLO(Sum)... succeeded.
      s 	w: 0
      #p^1_0	w: 0
      #p^1_s	w: 0
      p^1_0	w: 0
      rd	w: 0
      p^1_s	w: 0
      #half	w: 0
      #p 	w: 0
      #rd	w: x1
      f 	w: 0
      half	w: 0
      p 	w: 0
      0 	w: x1 + 1
      #f 	w: 0
      #g 	w: 0
      j^1_s	w: 0
      j 	w: 0
      #j^1_s	w: 0
      g 	w: 0
    USABLE RULES: { }
    Removed DPs: #7
Number of SCCs: 3, DPs: 8
  SCC { #12 #18 #19 }
POLO(Sum)... succeeded.
      s 	w: 0
      #p^1_0	w: x1 + 1
      #p^1_s	w: 0
      p^1_0	w: 0
      rd	w: 0
      p^1_s	w: x1 + 1
      #half	w: 0
      #p 	w: x1
      #rd	w: 0
      f 	w: 0
      half	w: 0
      p 	w: 0
      0 	w: x1 + 2
      #f 	w: 0
      #g 	w: 0
      j^1_s	w: 0
      j 	w: 0
      #j^1_s	w: 0
      g 	w: 0
    USABLE RULES: { }
    Removed DPs: #12 #18 #19
Number of SCCs: 2, DPs: 5
  SCC { #9 #20 }
POLO(Sum)... succeeded.
      s 	w: x1
      #p^1_0	w: 1
      #p^1_s	w: 0
      p^1_0	w: x1 + 1
      rd	w: 0
      p^1_s	w: x1
      #half	w: x1
      #p 	w: 0
      #rd	w: 0
      f 	w: 0
      half	w: 0
      p 	w: x1
      0 	w: x1 + 1
      #f 	w: 0
      #g 	w: 0
      j^1_s	w: 0
      j 	w: 0
      #j^1_s	w: 0
      g 	w: 0
    USABLE RULES: { 1..3 10 11 }
    Removed DPs: #9
Number of SCCs: 2, DPs: 4
  SCC { #20 }
POLO(Sum)... succeeded.
      s 	w: x1 + 26286
      #p^1_0	w: 1
      #p^1_s	w: 0
      p^1_0	w: 24867
      rd	w: 0
      p^1_s	w: x1 + 26285
      #half	w: x1
      #p 	w: 0
      #rd	w: 0
      f 	w: 0
      half	w: 0
      p 	w: x1
      0 	w: 24867
      #f 	w: 0
      #g 	w: 0
      j^1_s	w: 0
      j 	w: 0
      #j^1_s	w: 0
      g 	w: 0
    USABLE RULES: { 1..3 10 11 }
    Removed DPs: #20
Number of SCCs: 1, DPs: 3
  SCC { #3 #15 #24 }
POLO(Sum)... succeeded.
      s 	w: x1 + 239703
      #p^1_0	w: 1
      #p^1_s	w: 0
      p^1_0	w: 1
      rd	w: 0
      p^1_s	w: x1 + 47940
      #half	w: x1
      #p 	w: 0
      #rd	w: 0
      f 	w: 0
      half	w: 0
      p 	w: x1
      0 	w: 1
      #f 	w: x1 + 47941
      #g 	w: x1
      j^1_s	w: 0
      j 	w: 0
      #j^1_s	w: x1 + 143822
      g 	w: 0
    USABLE RULES: { 1..3 10 11 }
    Removed DPs: #3 #15 #24
Number of SCCs: 0, DPs: 0