Input TRS:
    1: plus(x,0()) -> x
    2: plus(x,s(y)) -> s(plus(x,y))
    3: times(0(),y) -> 0()
    4: times(x,0()) -> 0()
    5: times(s(x),y) -> plus(times(x,y),y)
    6: p(s(s(x))) -> s(p(s(x)))
    7: p(s(0())) -> 0()
    8: fac(s(x)) -> times(fac(p(s(x))),s(x))
Number of strict rules: 8
Direct Order(PosReal,>,Poly) ... failed.
Freezing p
1: plus(x,0()) -> x
2: plus(x,s(y)) -> s(plus(x,y))
3: times(0(),y) -> 0()
4: times(x,0()) -> 0()
5: times(s(x),y) -> plus(times(x,y),y)
6: p❆1_s(s(x)) -> s(p❆1_s(x))
7: p❆1_s(0()) -> 0()
8: fac(s(x)) -> times(fac(p❆1_s(x)),s(x))
9: p(s(_1)) ->= p❆1_s(_1)
Number of strict rules: 8
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #plus(x,s(y)) -> #plus(x,y)
   #2: #p❆1_s(s(x)) -> #p❆1_s(x)
   #3: #p(s(_1)) ->? #p❆1_s(_1)
   #4: #times(s(x),y) -> #plus(times(x,y),y)
   #5: #times(s(x),y) -> #times(x,y)
   #6: #fac(s(x)) -> #times(fac(p❆1_s(x)),s(x))
   #7: #fac(s(x)) -> #fac(p❆1_s(x))
   #8: #fac(s(x)) -> #p❆1_s(x)
Number of SCCs: 4, DPs: 4, edges: 4
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
 #fac(x1)	weight: 0
p❆1_s(x1)	weight: 0
#plus(x1,x2)	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: x1
     p(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
 times(x1,x2)	weight: 0
   fac(x1)	weight: 0
  plus(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 3, DPs: 3, edges: 3
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
 #fac(x1)	weight: x1
p❆1_s(x1)	weight: (/ 1 4) + x1
#plus(x1,x2)	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
 times(x1,x2)	weight: 0
   fac(x1)	weight: 0
  plus(x1,x2)	weight: 0
    Usable rules: { 6 7 }
    Removed DPs: #7
Number of SCCs: 2, DPs: 2, edges: 2
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
 #fac(x1)	weight: x1
p❆1_s(x1)	weight: (/ 1 4) + x1
#plus(x1,x2)	weight: x2
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
 times(x1,x2)	weight: 0
   fac(x1)	weight: 0
  plus(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #1
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
 #fac(x1)	weight: x1
p❆1_s(x1)	weight: (/ 1 4) + x1
#plus(x1,x2)	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
#times(x1,x2)	weight: x1
     0()	weight: 0
 times(x1,x2)	weight: 0
   fac(x1)	weight: 0
  plus(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #5
Number of SCCs: 0, DPs: 0, edges: 0
YES