Input TRS:
    1: p(s(x)) -> x
    2: fact(0()) -> s(0())
    3: fact(s(x)) -> *(s(x),fact(p(s(x))))
    4: *(0(),y) -> 0()
    5: *(s(x),y) -> +(*(x,y),y)
    6: +(x,0()) -> x
    7: +(x,s(y)) -> s(+(x,y))
Number of strict rules: 7
Direct Order(PosReal,>,Poly) ... failed.
Freezing * p
1: p❆1_s(x) -> x
2: fact(0()) -> s(0())
3: fact(s(x)) -> *❆1_s(x,fact(p❆1_s(x)))
4: *❆1_0(y) -> 0()
5: *❆1_s(x,y) -> +(*(x,y),y)
6: +(x,0()) -> x
7: +(x,s(y)) -> s(+(x,y))
8: p(s(_1)) ->= p❆1_s(_1)
9: *(0(),_1) ->= *❆1_0(_1)
10: *(s(_1),_2) ->= *❆1_s(_1,_2)
Number of strict rules: 7
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #*(0(),_1) ->? #*❆1_0(_1)
   #2: #+(x,s(y)) -> #+(x,y)
   #3: #*(s(_1),_2) ->? #*❆1_s(_1,_2)
   #4: #*❆1_s(x,y) -> #+(*(x,y),y)
   #5: #*❆1_s(x,y) -> #*(x,y)
   #6: #fact(s(x)) -> #*❆1_s(x,fact(p❆1_s(x)))
   #7: #fact(s(x)) -> #fact(p❆1_s(x))
   #8: #fact(s(x)) -> #p❆1_s(x)
   #9: #p(s(_1)) ->? #p❆1_s(_1)
Number of SCCs: 3, DPs: 4, edges: 4
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
*❆1_0(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_s(x1)	weight: (/ 1 4) + x1
   #*(x1,x2)	weight: 0
#fact(x1)	weight: x1
   #p(x1)	weight: 0
#*❆1_s(x1,x2)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
     0()	weight: 0
  fact(x1)	weight: 0
#*❆1_0(x1)	weight: 0
     +(x1,x2)	weight: 0
*❆1_s(x1,x2)	weight: 0
   #+(x1,x2)	weight: 0
     *(x1,x2)	weight: 0
    Usable rules: { 1 }
    Removed DPs: #7
Number of SCCs: 2, DPs: 3, edges: 3
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
*❆1_0(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_s(x1)	weight: (/ 1 4) + x1
   #*(x1,x2)	weight: 0
#fact(x1)	weight: x1
   #p(x1)	weight: 0
#*❆1_s(x1,x2)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
     0()	weight: 0
  fact(x1)	weight: 0
#*❆1_0(x1)	weight: 0
     +(x1,x2)	weight: 0
*❆1_s(x1,x2)	weight: 0
   #+(x1,x2)	weight: x2
     *(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #3 #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
*❆1_0(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_s(x1)	weight: (/ 1 4) + x1
   #*(x1,x2)	weight: x1
#fact(x1)	weight: x1
   #p(x1)	weight: 0
#*❆1_s(x1,x2)	weight: (/ 1 4) + x1
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
     0()	weight: 0
  fact(x1)	weight: 0
#*❆1_0(x1)	weight: 0
     +(x1,x2)	weight: 0
*❆1_s(x1,x2)	weight: 0
   #+(x1,x2)	weight: 0
     *(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #3 #5
Number of SCCs: 0, DPs: 0, edges: 0
YES