Input TRS:
    1: minus(x,0()) -> x
    2: minus(s(x),s(y)) -> minus(x,y)
    3: f(0()) -> s(0())
    4: f(s(x)) -> minus(s(x),g(f(x)))
    5: g(0()) -> 0()
    6: g(s(x)) -> minus(s(x),f(g(x)))
    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: #minus(s(x),s(y)) -> #minus(x,y)
   #2: #g(s(x)) -> #minus(s(x),f(g(x)))
   #3: #g(s(x)) -> #f(g(x))
   #4: #g(s(x)) -> #g(x)
   #5: #f(s(x)) -> #minus(s(x),g(f(x)))
   #6: #f(s(x)) -> #g(f(x))
   #7: #f(s(x)) -> #f(x)
Number of SCCs: 2, DPs: 5, edges: 9
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
     s(x1)	weight: x1	status: [x1]	precedence above: #minus
 minus(x1,x2)	weight: max{0, (/ 1 8) + x1}	status: []	precedence above: s #minus
     f(x1)	weight: (/ 1 4) + x1	status: [x1]	precedence above: s minus 0 #minus
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
     0()	weight: (/ 1 8)	status: []	precedence above:
   #f(x1)	weight: (/ 1 8)	status: []	precedence above:
   #g(x1)	weight: x1	status: []	precedence above:
#minus(x1,x2)	weight: x2	status: [x2]	precedence above: s
     g(x1)	weight: (/ 1 4) + x1	status: [x1]	precedence above: 0
    Removed DPs: #1
Number of SCCs: 1, DPs: 4, edges: 8
	SCC { #3 #4 #6 #7 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
     s(x1)	weight: x1	status: [x1]	precedence above: f 0 #minus g
 minus(x1,x2)	weight: max{0, x1}	status: x1
     f(x1)	weight: x1	status: [x1]	precedence above: s 0 #minus g
  rand(x1)	weight: (/ 1 2) + x1	status: []	precedence above:
     0()	weight: (/ 1 2)	status: []	precedence above:
   #f(x1)	weight: (/ 1 2) + x1	status: [x1]	precedence above: s f 0 #g #minus g
   #g(x1)	weight: (/ 1 2) + x1	status: [x1]	precedence above: s f 0 #f #minus g
#minus(x1,x2)	weight: x2	status: [x2]	precedence above:
     g(x1)	weight: x1	status: [x1]	precedence above: s f 0 #minus
    Removed DPs: #4 #7
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #3 #6 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
     s(x1)	weight: x1	status: [x1]	precedence above: f 0 #minus g
 minus(x1,x2)	weight: max{0, x1}	status: x1
     f(x1)	weight: x1	status: [x1]	precedence above: s 0 #minus g
  rand(x1)	weight: (/ 1 2) + x1	status: []	precedence above:
     0()	weight: (/ 1 2)	status: []	precedence above:
   #f(x1)	weight: (/ 1 2) + x1	status: [x1]	precedence above: s f 0 #g #minus g
   #g(x1)	weight: (/ 1 2) + x1	status: [x1]	precedence above: s f 0 #f #minus g
#minus(x1,x2)	weight: x2	status: [x2]	precedence above:
     g(x1)	weight: x1	status: [x1]	precedence above: 0 #minus
    Removed DPs: #3
Number of SCCs: 0, DPs: 0, edges: 0
YES