YES
Input TRS:
      1: plus(0(),x) -> x
      2: plus(s(x),y) -> s(plus(p(s(x)),y))
      3: times(0(),y) -> 0()
      4: times(s(x),y) -> plus(y,times(p(s(x)),y))
      5: p(s(0())) -> 0()
      6: p(s(s(x))) -> s(p(s(x)))
      7: fac(0(),x) -> x
      8: fac(s(x),y) -> fac(p(s(x)),times(s(x),y))
      9: factorial(x) -> fac(x,s(0()))
Number of strict rules: 9
Direct POLO(bPol) ... failed.
Uncurrying times p
      1: plus(0(),x) -> x
      2: plus(s(x),y) -> s(plus(p^1_s(x),y))
      3: times^1_0(y) -> 0()
      4: times^1_s(x,y) -> plus(y,times(p^1_s(x),y))
      5: p^1_s(0()) -> 0()
      6: p^1_s(s(x)) -> s(p^1_s(x))
      7: fac(0(),x) -> x
      8: fac(s(x),y) -> fac(p^1_s(x),times^1_s(x,y))
      9: factorial(x) -> fac(x,s(0()))
      10: p(s(_1)) ->= p^1_s(_1)
      11: times(0(),_1) ->= times^1_0(_1)
      12: times(s(_1),_2) ->= times^1_s(_1,_2)
Number of strict rules: 9
Direct POLO(bPol) ... failed.
Dependency Pairs:
     #1: #plus(s(x),y) -> #plus(p^1_s(x),y)
     #2: #plus(s(x),y) -> #p^1_s(x)
     #3: #p^1_s(s(x)) -> #p^1_s(x)
     #4: #factorial(x) -> #fac(x,s(0()))
     #5: #times(0(),_1) ->? #times^1_0(_1)
     #6: #times(s(_1),_2) ->? #times^1_s(_1,_2)
     #7: #p(s(_1)) ->? #p^1_s(_1)
     #8: #fac(s(x),y) -> #fac(p^1_s(x),times^1_s(x,y))
     #9: #fac(s(x),y) -> #p^1_s(x)
     #10: #fac(s(x),y) -> #times^1_s(x,y)
     #11: #times^1_s(x,y) -> #plus(y,times(p^1_s(x),y))
     #12: #times^1_s(x,y) -> #times(p^1_s(x),y)
     #13: #times^1_s(x,y) -> #p^1_s(x)
Number of SCCs: 4, DPs: 5
  SCC { #3 }
POLO(Sum)... succeeded.
      times^1_s	w: 0
      s 	w: x1 + 1
      #p^1_s	w: x1
      #fac	w: 0
      #plus	w: 0
      p^1_s	w: 0
      #p 	w: 0
      times^1_0	w: 0
      p 	w: 0
      #times	w: 0
      0 	w: 0
      times	w: 0
      fac	w: 0
      #times^1_0	w: 0
      plus	w: 0
      factorial	w: 0
      #factorial	w: 0
      #times^1_s	w: 0
    USABLE RULES: { }
    Removed DPs: #3
Number of SCCs: 3, DPs: 4
  SCC { #1 }
POLO(Sum)... succeeded.
      times^1_s	w: 0
      s 	w: x1 + 2
      #p^1_s	w: 0
      #fac	w: 0
      #plus	w: x1
      p^1_s	w: x1 + 1
      #p 	w: 0
      times^1_0	w: 0
      p 	w: 0
      #times	w: 0
      0 	w: 23676
      times	w: 0
      fac	w: 0
      #times^1_0	w: 0
      plus	w: 0
      factorial	w: 0
      #factorial	w: 0
      #times^1_s	w: 0
    USABLE RULES: { 5 6 }
    Removed DPs: #1
Number of SCCs: 2, DPs: 3
  SCC { #8 }
POLO(Sum)... succeeded.
      times^1_s	w: 3
      s 	w: x1 + 2
      #p^1_s	w: 0
      #fac	w: x1
      #plus	w: x1
      p^1_s	w: x1 + 1
      #p 	w: 0
      times^1_0	w: x1 + 8945
      p 	w: 0
      #times	w: 0
      0 	w: 8946
      times	w: x1
      fac	w: 0
      #times^1_0	w: 0
      plus	w: 4
      factorial	w: 0
      #factorial	w: 0
      #times^1_s	w: 0
    USABLE RULES: { 5 6 }
    Removed DPs: #8
Number of SCCs: 1, DPs: 2
  SCC { #6 #12 }
POLO(Sum)... succeeded.
      times^1_s	w: 4
      s 	w: x1 + 3
      #p^1_s	w: 0
      #fac	w: x1
      #plus	w: x1
      p^1_s	w: x1 + 1
      #p 	w: 0
      times^1_0	w: x1 + 8945
      p 	w: 0
      #times	w: x1
      0 	w: 8946
      times	w: x1
      fac	w: 0
      #times^1_0	w: 0
      plus	w: 5
      factorial	w: 0
      #factorial	w: 0
      #times^1_s	w: x1 + 2
    USABLE RULES: { 5 6 }
    Removed DPs: #6 #12
Number of SCCs: 0, DPs: 0