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