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