Input TRS:
    1: +(|0|(),y) -> y
    2: +(s(x),y) -> s(+(x,y))
    3: ++(nil(),ys) -> ys
    4: ++(|:|(x,xs),ys) -> |:|(x,++(xs,ys))
    5: sum(|:|(x,nil())) -> |:|(x,nil())
    6: sum(|:|(x,|:|(y,xs))) -> sum(|:|(+(x,y),xs))
    7: sum(++(xs,|:|(x,|:|(y,ys)))) -> sum(++(xs,sum(|:|(x,|:|(y,ys)))))
    8: -(x,|0|()) -> x
    9: -(|0|(),s(y)) -> |0|()
    10: -(s(x),s(y)) -> -(x,y)
    11: quot(|0|(),s(y)) -> |0|()
    12: quot(s(x),s(y)) -> s(quot(-(x,y),s(y)))
    13: length(nil()) -> |0|()
    14: length(|:|(x,xs)) -> s(length(xs))
    15: hd(|:|(x,xs)) -> x
    16: avg(xs) -> quot(hd(sum(xs)),length(xs))
Number of strict rules: 16
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #+(s(x),y) -> #+(x,y)
   #2: #sum(|:|(x,|:|(y,xs))) -> #sum(|:|(+(x,y),xs))
   #3: #sum(|:|(x,|:|(y,xs))) -> #+(x,y)
   #4: #quot(s(x),s(y)) -> #quot(-(x,y),s(y))
   #5: #quot(s(x),s(y)) -> #-(x,y)
   #6: #length(|:|(x,xs)) -> #length(xs)
   #7: #sum(++(xs,|:|(x,|:|(y,ys)))) -> #sum(++(xs,sum(|:|(x,|:|(y,ys)))))
   #8: #sum(++(xs,|:|(x,|:|(y,ys)))) -> #++(xs,sum(|:|(x,|:|(y,ys))))
   #9: #sum(++(xs,|:|(x,|:|(y,ys)))) -> #sum(|:|(x,|:|(y,ys)))
   #10: #-(s(x),s(y)) -> #-(x,y)
   #11: #avg(xs) -> #quot(hd(sum(xs)),length(xs))
   #12: #avg(xs) -> #hd(sum(xs))
   #13: #avg(xs) -> #sum(xs)
   #14: #avg(xs) -> #length(xs)
   #15: #++(|:|(x,xs),ys) -> #++(xs,ys)
Number of SCCs: 7, DPs: 7, edges: 7
	SCC { #6 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
    hd(x1)	weight: 0
     s(x1)	weight: 0
    ++(x1,x2)	weight: 0
   sum(x1)	weight: 0
 #avg(x1)	weight: 0
  quot(x1,x2)	weight: 0
  #++(x1,x2)	weight: 0
   nil()	weight: 0
     -(x1,x2)	weight: 0
   avg(x1)	weight: 0
  #hd(x1)	weight: 0
   #-(x1,x2)	weight: 0
   |:|(x1,x2)	weight: (/ 1 2) + x2
#quot(x1,x2)	weight: 0
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: x1
   #+(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #6
Number of SCCs: 6, DPs: 6, edges: 6
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
    hd(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
    ++(x1,x2)	weight: 0
   sum(x1)	weight: 0
 #avg(x1)	weight: 0
  quot(x1,x2)	weight: 0
  #++(x1,x2)	weight: 0
   nil()	weight: 0
     -(x1,x2)	weight: 0
   avg(x1)	weight: 0
  #hd(x1)	weight: 0
   #-(x1,x2)	weight: 0
   |:|(x1,x2)	weight: (/ 1 2)
#quot(x1,x2)	weight: 0
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: 0
   #+(x1,x2)	weight: x1
    Usable rules: { }
    Removed DPs: #1
Number of SCCs: 5, DPs: 5, edges: 5
	SCC { #15 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
    hd(x1)	weight: 0
     s(x1)	weight: (/ 1 2)
    ++(x1,x2)	weight: 0
   sum(x1)	weight: 0
 #avg(x1)	weight: 0
  quot(x1,x2)	weight: 0
  #++(x1,x2)	weight: x1
   nil()	weight: 0
     -(x1,x2)	weight: 0
   avg(x1)	weight: 0
  #hd(x1)	weight: 0
   #-(x1,x2)	weight: 0
   |:|(x1,x2)	weight: (/ 1 2) + x2
#quot(x1,x2)	weight: 0
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: 0
   #+(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #15
Number of SCCs: 4, DPs: 4, edges: 4
	SCC { #10 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
    hd(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
    ++(x1,x2)	weight: 0
   sum(x1)	weight: 0
 #avg(x1)	weight: 0
  quot(x1,x2)	weight: 0
  #++(x1,x2)	weight: 0
   nil()	weight: 0
     -(x1,x2)	weight: 0
   avg(x1)	weight: 0
  #hd(x1)	weight: 0
   #-(x1,x2)	weight: x2
   |:|(x1,x2)	weight: (/ 1 2)
#quot(x1,x2)	weight: 0
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: 0
   #+(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #10
Number of SCCs: 3, DPs: 3, edges: 3
	SCC { #4 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
    hd(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
    ++(x1,x2)	weight: 0
   sum(x1)	weight: 0
 #avg(x1)	weight: 0
  quot(x1,x2)	weight: 0
  #++(x1,x2)	weight: 0
   nil()	weight: 0
     -(x1,x2)	weight: (/ 1 4) + x1
   avg(x1)	weight: 0
  #hd(x1)	weight: 0
   #-(x1,x2)	weight: 0
   |:|(x1,x2)	weight: (/ 1 4)
#quot(x1,x2)	weight: x1
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
length(x1)	weight: 0
#length(x1)	weight: 0
   #+(x1,x2)	weight: 0
    Usable rules: { 8..10 }
    Removed DPs: #4
Number of SCCs: 2, DPs: 2, edges: 2
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
    hd(x1)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
    ++(x1,x2)	weight: 0
   sum(x1)	weight: 0
 #avg(x1)	weight: 0
  quot(x1,x2)	weight: 0
  #++(x1,x2)	weight: 0
   nil()	weight: 0
     -(x1,x2)	weight: (/ 1 4) + x1
   avg(x1)	weight: 0
  #hd(x1)	weight: 0
   #-(x1,x2)	weight: 0
   |:|(x1,x2)	weight: (/ 1 4) + x2
#quot(x1,x2)	weight: x1
     +(x1,x2)	weight: (/ 1 4)
 #sum(x1)	weight: x1
length(x1)	weight: 0
#length(x1)	weight: 0
   #+(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
    hd(x1)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
    ++(x1,x2)	weight: (/ 1 4) + x1 + x2
   sum(x1)	weight: (/ 1 4)
 #avg(x1)	weight: 0
  quot(x1,x2)	weight: 0
  #++(x1,x2)	weight: 0
   nil()	weight: 0
     -(x1,x2)	weight: (/ 1 4) + x1
   avg(x1)	weight: 0
  #hd(x1)	weight: 0
   #-(x1,x2)	weight: 0
   |:|(x1,x2)	weight: (/ 1 4) + x2
#quot(x1,x2)	weight: x1
     +(x1,x2)	weight: (/ 1 4)
 #sum(x1)	weight: x1
length(x1)	weight: 0
#length(x1)	weight: 0
   #+(x1,x2)	weight: 0
    Usable rules: { 3..6 }
    Removed DPs: #7
Number of SCCs: 0, DPs: 0, edges: 0
YES