Input TRS:
    1: p(s(x)) -> x
    2: fac(0()) -> s(0())
    3: fac(s(x)) -> times(s(x),fac(p(s(x))))
    4: rand(x) ->= x
    5: rand(x) ->= rand(s(x))
Number of strict rules: 3
Direct Order(PosReal,>,Poly) ... failed.
Freezing p
1: p❆1_s(x) -> x
2: fac(0()) -> s(0())
3: fac(s(x)) -> times(s(x),fac(p❆1_s(x)))
4: rand(x) ->= x
5: rand(x) ->= rand(s(x))
6: p(s(_1)) ->= p❆1_s(_1)
Number of strict rules: 3
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #p(s(_1)) ->? #p❆1_s(_1)
   #2: #fac(s(x)) -> #fac(p❆1_s(x))
   #3: #fac(s(x)) -> #p❆1_s(x)
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
     s(x1)	weight: x1	status: [x1]	precedence above: p❆1_s p
 #fac(x1)	weight: x1	status: [x1]	precedence above:
p❆1_s(x1)	weight: x1	status: [x1]	precedence above: p
   #p(x1)	weight: x1	status: []	precedence above:
#p❆1_s(x1)	weight: x1	status: []	precedence above:
  rand(x1)	weight: (/ 1 4) + x1	status: []	precedence above:
     p(x1)	weight: (/ 1 4) + x1	status: []	precedence above:
     0()	weight: (/ 1 4)	status: []	precedence above:
 times(x1,x2)	weight: 0	status: []	precedence above: s p❆1_s p 0 fac
   fac(x1)	weight: (/ 1 2)	status: []	precedence above: s p❆1_s p 0
    Removed DPs: #2
Number of SCCs: 0, DPs: 0, edges: 0
YES