Input TRS:
    1: nonZero(0()) -> false()
    2: nonZero(s(x)) -> true()
    3: p(s(0())) -> 0()
    4: p(s(s(x))) -> s(p(s(x)))
    5: id_inc(x) -> x
    6: id_inc(x) -> s(x)
    7: random(x) -> rand(x,0())
    8: rand(x,y) -> if(nonZero(x),x,y)
    9: if(false(),x,y) -> y
    10: if(true(),x,y) -> rand(p(x),id_inc(y))
Number of strict rules: 10
Direct Order(PosReal,>,Poly) ... failed.
Freezing p
1: nonZero(0()) -> false()
2: nonZero(s(x)) -> true()
3: p❆1_s(0()) -> 0()
4: p❆1_s(s(x)) -> s(p❆1_s(x))
5: id_inc(x) -> x
6: id_inc(x) -> s(x)
7: random(x) -> rand(x,0())
8: rand(x,y) -> if(nonZero(x),x,y)
9: if(false(),x,y) -> y
10: if(true(),x,y) -> rand(p(x),id_inc(y))
11: p(s(_1)) ->= p❆1_s(_1)
Number of strict rules: 10
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #p(s(_1)) ->? #p❆1_s(_1)
   #2: #random(x) -> #rand(x,0())
   #3: #if(true(),x,y) -> #rand(p(x),id_inc(y))
   #4: #if(true(),x,y) -> #p(x)
   #5: #if(true(),x,y) -> #id_inc(y)
   #6: #rand(x,y) -> #if(nonZero(x),x,y)
   #7: #rand(x,y) -> #nonZero(x)
   #8: #p❆1_s(s(x)) -> #p❆1_s(x)
Number of SCCs: 2, DPs: 3, edges: 3
	SCC { #8 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
#rand(x1,x2)	weight: 0
nonZero(x1)	weight: 0
p❆1_s(x1)	weight: 0
 false()	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: x1
  true()	weight: 0
#nonZero(x1)	weight: 0
  rand(x1,x2)	weight: 0
#id_inc(x1)	weight: 0
     p(x1)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
random(x1)	weight: 0
#random(x1)	weight: 0
id_inc(x1)	weight: 0
  #if(x1,x2,x3)	weight: 0
    Usable rules: { }
    Removed DPs: #8
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #3 #6 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... succeeded.
     s(x1)	weight: max{0, (/ 1 2) + x1}
#rand(x1,x2)	weight: max{0, (/ 1 4) + x1}
nonZero(x1)	weight: max{0, x1}
p❆1_s(x1)	weight: max{0, (/ 1 8) + x1}
 false()	weight: (/ 1 8)
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
  true()	weight: (/ 1 8)
#nonZero(x1)	weight: 0
  rand(x1,x2)	weight: 0
#id_inc(x1)	weight: 0
     p(x1)	weight: max{0, (- (/ 3 8)) + x1}
     0()	weight: (/ 1 8)
    if(x1,x2,x3)	weight: 0
random(x1)	weight: 0
#random(x1)	weight: 0
id_inc(x1)	weight: (max (/ 1 4) 0)
  #if(x1,x2,x3)	weight: max{0, (/ 1 4) + x1, (/ 1 8) + x2}
    Usable rules: { 1..4 11 }
    Removed DPs: #3
Number of SCCs: 0, DPs: 0, edges: 0
YES