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: le(|0|(),y) -> true()
    6: le(s(x),|0|()) -> false()
    7: le(s(x),s(y)) -> le(x,y)
    8: app(nil(),y) -> y
    9: app(add(n,x),y) -> add(n,app(x,y))
    10: low(n,nil()) -> nil()
    11: low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
    12: if_low(true(),n,add(m,x)) -> add(m,low(n,x))
    13: if_low(false(),n,add(m,x)) -> low(n,x)
    14: high(n,nil()) -> nil()
    15: high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
    16: if_high(true(),n,add(m,x)) -> high(n,x)
    17: if_high(false(),n,add(m,x)) -> add(m,high(n,x))
    18: quicksort(nil()) -> nil()
    19: quicksort(add(n,x)) -> app(quicksort(low(n,x)),add(n,quicksort(high(n,x))))
    20: rand(x) ->= x
    21: rand(x) ->= rand(s(x))
Number of strict rules: 19
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #minus(s(x),s(y)) -> #minus(x,y)
   #2: #if_low(false(),n,add(m,x)) -> #low(n,x)
   #3: #app(add(n,x),y) -> #app(x,y)
   #4: #low(n,add(m,x)) -> #if_low(le(m,n),n,add(m,x))
   #5: #low(n,add(m,x)) -> #le(m,n)
   #6: #if_low(true(),n,add(m,x)) -> #low(n,x)
   #7: #le(s(x),s(y)) -> #le(x,y)
   #8: #if_high(false(),n,add(m,x)) -> #high(n,x)
   #9: #quicksort(add(n,x)) -> #app(quicksort(low(n,x)),add(n,quicksort(high(n,x))))
   #10: #quicksort(add(n,x)) -> #quicksort(low(n,x))
   #11: #quicksort(add(n,x)) -> #low(n,x)
   #12: #quicksort(add(n,x)) -> #quicksort(high(n,x))
   #13: #quicksort(add(n,x)) -> #high(n,x)
   #14: #if_high(true(),n,add(m,x)) -> #high(n,x)
   #15: #high(n,add(m,x)) -> #if_high(le(m,n),n,add(m,x))
   #16: #high(n,add(m,x)) -> #le(m,n)
   #17: #quot(s(x),s(y)) -> #quot(minus(x,y),s(y))
   #18: #quot(s(x),s(y)) -> #minus(x,y)
Number of SCCs: 7, DPs: 12, edges: 16
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
   |0|()	weight: (/ 1 4)	status: []	precedence above: false
    le(x1,x2)	weight: max{0, (/ 1 2) + x2}	status: [x2]	precedence above: false true
     s(x1)	weight: x1	status: [x1]	precedence above: #le quot
  #le(x1,x2)	weight: x2	status: [x2]	precedence above: s quot
 minus(x1,x2)	weight: max{0, x1}	status: x1
if_high(x1,x2,x3)	weight: 0	status: []	precedence above: le false quicksort true high nil low add if_low app
#quicksort(x1)	weight: x1	status: []	precedence above:
#high(x1,x2)	weight: max{0, (/ 1 8) + x2}	status: [x2]	precedence above:
 false()	weight: (/ 1 8)	status: []	precedence above:
#if_high(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x2,x1,x3]	precedence above:
quicksort(x1)	weight: (/ 1 4)	status: []	precedence above: le false true nil low add if_low app
  true()	weight: (/ 3 8)	status: []	precedence above:
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
  quot(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above: s #le
  high(x1,x2)	weight: 0	status: []	precedence above: le if_high false quicksort true nil low add if_low app
   nil()	weight: 0	status: []	precedence above:
 #app(x1,x2)	weight: (/ 1 8) + x2 + x1	status: [x2,x1]	precedence above:
#minus(x1,x2)	weight: (/ 1 8) + x2 + x1	status: [x1,x2]	precedence above:
   low(x1,x2)	weight: max{0, (/ 5 8) + x1}	status: [x1]	precedence above: le false quicksort true nil add if_low app
   add(x1,x2)	weight: 0	status: []	precedence above: le false quicksort true nil low if_low app
#quot(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above:
if_low(x1,x2,x3)	weight: max{0, (/ 5 8) + x2, x1}	status: [x2]	precedence above: le false quicksort true nil low add app
#if_low(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x1,x2,x3]	precedence above:
 #low(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above:
   app(x1,x2)	weight: max{0, (/ 1 8) + x2}	status: []	precedence above: le false quicksort true nil low add if_low
    Removed DPs: #7
Number of SCCs: 6, DPs: 11, edges: 15
	SCC { #3 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   |0|()	status: []	precedence above: false
    le(x1,x2)	status: []	precedence above: false true nil #app
     s(x1)	status: x1
  #le(x1,x2)	status: []	precedence above:
 minus(x1,x2)	status: [x1]	precedence above: |0| s false quot
if_high(x1,x2,x3)	status: [x3]	precedence above: le false true high nil #app low add if_low
#quicksort(x1)	status: []	precedence above:
#high(x1,x2)	status: [x1]	precedence above:
 false()	status: []	precedence above:
#if_high(x1,x2,x3)	status: [x1,x3,x2]	precedence above:
quicksort(x1)	status: [x1]	precedence above: le if_high false true high nil #app low add if_low app
  true()	status: []	precedence above:
  rand(x1)	status: [x1]	precedence above:
  quot(x1,x2)	status: []	precedence above: |0| s false
  high(x1,x2)	status: [x2]	precedence above: le if_high false true nil #app low add if_low
   nil()	status: []	precedence above:
 #app(x1,x2)	status: [x1]	precedence above:
#minus(x1,x2)	status: [x2]	precedence above:
   low(x1,x2)	status: [x2]	precedence above: le if_high false true high nil #app add if_low
   add(x1,x2)	status: [x2,x1]	precedence above: le if_high false true high nil #app low if_low
#quot(x1,x2)	status: [x1,x2]	precedence above:
if_low(x1,x2,x3)	status: [x3]	precedence above: le if_high false true high nil #app low add
#if_low(x1,x2,x3)	status: []	precedence above:
 #low(x1,x2)	status: [x2]	precedence above:
   app(x1,x2)	status: [x2,x1]	precedence above: le if_high false true high nil #app low add if_low
    Removed DPs: #3
Number of SCCs: 5, DPs: 10, edges: 14
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
   |0|()	weight: (/ 1 4)	status: []	precedence above: false
    le(x1,x2)	weight: max{0, (/ 1 2) + x2}	status: [x2]	precedence above: false true
     s(x1)	weight: x1	status: [x1]	precedence above: #le quot
  #le(x1,x2)	weight: x2	status: [x2]	precedence above: s quot
 minus(x1,x2)	weight: max{0, x1}	status: x1
if_high(x1,x2,x3)	weight: 0	status: []	precedence above: le false quicksort true high nil low add if_low app
#quicksort(x1)	weight: x1	status: []	precedence above:
#high(x1,x2)	weight: max{0, (/ 1 8) + x2}	status: [x2]	precedence above:
 false()	weight: (/ 1 8)	status: []	precedence above:
#if_high(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x2,x1,x3]	precedence above:
quicksort(x1)	weight: (/ 1 4)	status: []	precedence above: le false true nil low add if_low app
  true()	weight: (/ 3 8)	status: []	precedence above:
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
  quot(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above: s #le
  high(x1,x2)	weight: 0	status: []	precedence above: le if_high false quicksort true nil low add if_low app
   nil()	weight: 0	status: []	precedence above:
 #app(x1,x2)	weight: (/ 1 8) + x2 + x1	status: [x2,x1]	precedence above:
#minus(x1,x2)	weight: x2	status: [x2]	precedence above:
   low(x1,x2)	weight: max{0, (/ 5 8) + x1}	status: [x1]	precedence above: le false quicksort true nil add if_low app
   add(x1,x2)	weight: 0	status: []	precedence above: le false quicksort true nil low if_low app
#quot(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above:
if_low(x1,x2,x3)	weight: max{0, (/ 5 8) + x2, x1}	status: [x2]	precedence above: le false quicksort true nil low add app
#if_low(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x1,x2,x3]	precedence above:
 #low(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above:
   app(x1,x2)	weight: max{0, (/ 1 8) + x2}	status: []	precedence above: le false quicksort true nil low add if_low
    Removed DPs: #1
Number of SCCs: 4, DPs: 9, edges: 13
	SCC { #17 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
   |0|()	weight: (/ 1 4)	status: []	precedence above: false
    le(x1,x2)	weight: max{0, (/ 1 2) + x2}	status: [x2]	precedence above: false true
     s(x1)	weight: x1	status: [x1]	precedence above: #le minus quot
  #le(x1,x2)	weight: x2	status: [x2]	precedence above: s minus quot
 minus(x1,x2)	weight: max{0, x1}	status: x1
if_high(x1,x2,x3)	weight: 0	status: []	precedence above: le false quicksort true high nil low add if_low app
#quicksort(x1)	weight: x1	status: []	precedence above:
#high(x1,x2)	weight: max{0, (/ 1 8) + x2}	status: [x2]	precedence above:
 false()	weight: (/ 1 8)	status: []	precedence above:
#if_high(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x2,x1,x3]	precedence above:
quicksort(x1)	weight: (/ 1 4)	status: []	precedence above: le false true nil low add if_low app
  true()	weight: (/ 3 8)	status: []	precedence above:
  rand(x1)	weight: (/ 1 8) + x1	status: []	precedence above:
  quot(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above: s #le minus
  high(x1,x2)	weight: 0	status: []	precedence above: le if_high false quicksort true nil low add if_low app
   nil()	weight: 0	status: []	precedence above:
 #app(x1,x2)	weight: (/ 1 8) + x2 + x1	status: [x2,x1]	precedence above:
#minus(x1,x2)	weight: x2	status: [x2]	precedence above:
   low(x1,x2)	weight: max{0, (/ 5 8) + x1}	status: [x1]	precedence above: le false quicksort true nil add if_low app
   add(x1,x2)	weight: 0	status: []	precedence above: le false quicksort true nil low if_low app
#quot(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 8) + x1}	status: [x2,x1]	precedence above: s #le minus quot
if_low(x1,x2,x3)	weight: max{0, (/ 5 8) + x2, x1}	status: [x2]	precedence above: le false quicksort true nil low add app
#if_low(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 8) + x2, (/ 1 8) + x1}	status: [x1,x2,x3]	precedence above:
 #low(x1,x2)	weight: max{0, x1}	status: [x1]	precedence above:
   app(x1,x2)	weight: max{0, (/ 1 8) + x2}	status: []	precedence above: le false quicksort true nil low add if_low
    Removed DPs: #17
Number of SCCs: 3, DPs: 8, edges: 12
	SCC { #10 #12 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   |0|()	status: []	precedence above: false
    le(x1,x2)	status: []	precedence above: false true nil #app
     s(x1)	status: x1
  #le(x1,x2)	status: []	precedence above:
 minus(x1,x2)	status: x1
if_high(x1,x2,x3)	status: [x3]	precedence above: le false true high nil #app low add if_low
#quicksort(x1)	status: [x1]	precedence above:
#high(x1,x2)	status: [x1]	precedence above:
 false()	status: []	precedence above:
#if_high(x1,x2,x3)	status: [x1,x3,x2]	precedence above:
quicksort(x1)	status: [x1]	precedence above: le if_high false true high nil #app low add if_low app
  true()	status: []	precedence above:
  rand(x1)	status: [x1]	precedence above:
  quot(x1,x2)	status: x1
  high(x1,x2)	status: [x2]	precedence above: le if_high false true nil #app low add if_low
   nil()	status: []	precedence above:
 #app(x1,x2)	status: [x1]	precedence above:
#minus(x1,x2)	status: []	precedence above:
   low(x1,x2)	status: [x2]	precedence above: le if_high false true high nil #app add if_low
   add(x1,x2)	status: [x2,x1]	precedence above: le if_high false true high nil #app low if_low
#quot(x1,x2)	status: []	precedence above:
if_low(x1,x2,x3)	status: [x3]	precedence above: le if_high false true high nil #app low add
#if_low(x1,x2,x3)	status: []	precedence above:
 #low(x1,x2)	status: [x2]	precedence above:
   app(x1,x2)	status: [x2,x1]	precedence above: le if_high false true high nil #app low add if_low
    Removed DPs: #10 #12
Number of SCCs: 2, DPs: 6, edges: 8
	SCC { #2 #4 #6 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   |0|()	status: []	precedence above: false
    le(x1,x2)	status: []	precedence above: false true nil #app
     s(x1)	status: x1
  #le(x1,x2)	status: []	precedence above:
 minus(x1,x2)	status: x1
if_high(x1,x2,x3)	status: [x3]	precedence above: le false true high nil #app low add if_low #if_low #low
#quicksort(x1)	status: [x1]	precedence above:
#high(x1,x2)	status: [x1]	precedence above:
 false()	status: []	precedence above:
#if_high(x1,x2,x3)	status: [x1,x3,x2]	precedence above:
quicksort(x1)	status: [x1]	precedence above: le if_high false true high nil #app low add if_low #if_low #low app
  true()	status: []	precedence above:
  rand(x1)	status: [x1]	precedence above:
  quot(x1,x2)	status: x1
  high(x1,x2)	status: [x2]	precedence above: le if_high false true nil #app low add if_low #if_low #low
   nil()	status: []	precedence above:
 #app(x1,x2)	status: [x1]	precedence above:
#minus(x1,x2)	status: []	precedence above:
   low(x1,x2)	status: [x2]	precedence above: le if_high false true high nil #app add if_low #if_low #low
   add(x1,x2)	status: [x2,x1]	precedence above: le if_high false true high nil #app low if_low #if_low #low
#quot(x1,x2)	status: []	precedence above:
if_low(x1,x2,x3)	status: [x3]	precedence above: le if_high false true high nil #app low add #if_low #low
#if_low(x1,x2,x3)	status: [x3]	precedence above: #low
 #low(x1,x2)	status: [x2]	precedence above: #if_low
   app(x1,x2)	status: [x2,x1]	precedence above: le if_high false true high nil #app low add if_low #if_low #low
    Removed DPs: #2 #6
Number of SCCs: 1, DPs: 3, edges: 4
	SCC { #8 #14 #15 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   |0|()	status: []	precedence above: false
    le(x1,x2)	status: []	precedence above: false true nil #app
     s(x1)	status: x1
  #le(x1,x2)	status: []	precedence above:
 minus(x1,x2)	status: x1
if_high(x1,x2,x3)	status: [x3]	precedence above: le #high false true high nil #app low add if_low #if_low #low
#quicksort(x1)	status: [x1]	precedence above:
#high(x1,x2)	status: [x2]	precedence above: le if_high false true high nil #app low add if_low #if_low #low
 false()	status: []	precedence above:
#if_high(x1,x2,x3)	status: x3
quicksort(x1)	status: [x1]	precedence above: le if_high #high false true high nil #app low add if_low #if_low #low app
  true()	status: []	precedence above:
  rand(x1)	status: [x1]	precedence above:
  quot(x1,x2)	status: x1
  high(x1,x2)	status: [x2]	precedence above: le if_high #high false true nil #app low add if_low #if_low #low
   nil()	status: []	precedence above:
 #app(x1,x2)	status: [x1]	precedence above:
#minus(x1,x2)	status: []	precedence above:
   low(x1,x2)	status: [x2]	precedence above: le if_high #high false true high nil #app add if_low #if_low #low
   add(x1,x2)	status: [x2,x1]	precedence above: le if_high #high false true high nil #app low if_low #if_low #low
#quot(x1,x2)	status: []	precedence above:
if_low(x1,x2,x3)	status: [x3]	precedence above: le if_high #high false true high nil #app low add #if_low #low
#if_low(x1,x2,x3)	status: [x3]	precedence above: #low
 #low(x1,x2)	status: [x2]	precedence above: #if_low
   app(x1,x2)	status: [x2,x1]	precedence above: le if_high #high false true high nil #app low add if_low #if_low #low
    Removed DPs: #8 #14 #15
Number of SCCs: 0, DPs: 0, edges: 0
YES