YES
Input TRS:
      1: log(s(x1)) -> s(log(half(s(x1))))
      2: half(0(x1)) -> 0(s(s(half(x1))))
      3: half(s(0(x1))) -> 0(x1)
      4: half(s(s(x1))) -> s(half(p(s(s(x1)))))
      5: half(half(s(s(s(s(x1)))))) -> s(s(half(half(x1))))
      6: p(s(s(s(x1)))) -> s(p(s(s(x1))))
      7: s(s(p(s(x1)))) -> s(s(x1))
      8: 0(x1) -> x1
Number of strict rules: 8
Direct POLO(bPol) ... removes: 8
      s 	w: x1
      half	w: x1
      p 	w: x1
      log	w: x1 + 282
      0 	w: x1 + 20538
Number of strict rules: 7
Direct POLO(bPol) ... failed.
Uncurrying ... failed.
Dependency Pairs:
     #1: #half(0(x1)) -> #s(s(half(x1)))
     #2: #half(0(x1)) -> #s(half(x1))
     #3: #half(0(x1)) -> #half(x1)
     #4: #p(s(s(s(x1)))) -> #s(p(s(s(x1))))
     #5: #p(s(s(s(x1)))) -> #p(s(s(x1)))
     #6: #s(s(p(s(x1)))) -> #s(s(x1))
     #7: #half(half(s(s(s(s(x1)))))) -> #s(s(half(half(x1))))
     #8: #half(half(s(s(s(s(x1)))))) -> #s(half(half(x1)))
     #9: #half(half(s(s(s(s(x1)))))) -> #half(half(x1))
     #10: #half(half(s(s(s(s(x1)))))) -> #half(x1)
     #11: #log(s(x1)) -> #s(log(half(s(x1))))
     #12: #log(s(x1)) -> #log(half(s(x1)))
     #13: #log(s(x1)) -> #half(s(x1))
     #14: #half(s(s(x1))) -> #s(half(p(s(s(x1)))))
     #15: #half(s(s(x1))) -> #half(p(s(s(x1))))
     #16: #half(s(s(x1))) -> #p(s(s(x1)))
Number of SCCs: 5, DPs: 7
  SCC { #12 }
POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... succeeded.
      s 	w: max(x1 + 20165, 0)
      #log	w: max(x1 - 20164, 0)
      #half	w: 0
      #p 	w: 0
      half	w: max(x1 - 11799, 0)
      p 	w: max(x1 - 20165, 0)
      log	w: 0
      0 	w: 1
      #s 	w: max(x1 - 1, 0)
    USABLE RULES: { 3 4 6 7 }
    Removed DPs: #12
Number of SCCs: 4, DPs: 6
  SCC { #6 }
POLO(Sum)... succeeded.
      s 	w: x1 + 1
      #log	w: 0
      #half	w: 0
      #p 	w: 0
      half	w: 41063
      p 	w: x1
      log	w: 0
      0 	w: 1
      #s 	w: x1
    USABLE RULES: { 3 7 }
    Removed DPs: #6
Number of SCCs: 3, DPs: 5
  SCC { #15 }
POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... succeeded.
      s 	w: max(x1 + 30614, 0)
      #log	w: max(x1 - 20164, 0)
      #half	w: max(x1 - 1, 0)
      #p 	w: 0
      half	w: max(x1 - 1, 0)
      p 	w: max(x1 - 30614, 0)
      log	w: 0
      0 	w: 1
      #s 	w: max(x1 - 1, 0)
    USABLE RULES: { 3 4 6 7 }
    Removed DPs: #15
Number of SCCs: 2, DPs: 4
  SCC { #5 }
POLO(Sum)... succeeded.
      s 	w: x1 + 1
      #log	w: 0
      #half	w: 0
      #p 	w: x1
      half	w: 35231
      p 	w: x1 + 15921
      log	w: 0
      0 	w: 2998
      #s 	w: 0
    USABLE RULES: { 7 }
    Removed DPs: #5
Number of SCCs: 1, DPs: 3
  SCC { #3 #9 #10 }
POLO(Sum)... succeeded.
      s 	w: x1
      #log	w: 0
      #half	w: x1
      #p 	w: x1
      half	w: x1 + 19781
      p 	w: x1
      log	w: 0
      0 	w: x1 + 25907
      #s 	w: 0
    USABLE RULES: { 2..7 }
    Removed DPs: #3 #10
Number of SCCs: 1, DPs: 1
  SCC { #9 }
POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... succeeded.
      s 	w: max(x1 + 1709, 0)
      #log	w: max(x1 - 20164, 0)
      #half	w: max(x1 - 1, 0)
      #p 	w: 0
      half	w: max(x1 - 1426, 0)
      p 	w: max(x1 - 1709, 0)
      log	w: 0
      0 	w: 0
      #s 	w: max(x1 - 1, 0)
    USABLE RULES: { 2..7 }
    Removed DPs: #9
Number of SCCs: 0, DPs: 0