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