Input TRS:
    1: |0|(|\#|()) -> |\#|()
    2: +(x,|\#|()) -> x
    3: +(|\#|(),x) -> x
    4: +(|0|(x),|0|(y)) -> |0|(+(x,y))
    5: +(|0|(x),|1|(y)) -> |1|(+(x,y))
    6: +(|1|(x),|0|(y)) -> |1|(+(x,y))
    7: +(|1|(x),|1|(y)) -> |0|(+(+(x,y),|1|(|\#|())))
    8: +(+(x,y),z) -> +(x,+(y,z))
    9: *(|\#|(),x) -> |\#|()
    10: *(|0|(x),y) -> |0|(*(x,y))
    11: *(|1|(x),y) -> +(|0|(*(x,y)),y)
    12: *(*(x,y),z) -> *(x,*(y,z))
    13: sum(nil()) -> |0|(|\#|())
    14: sum(cons(x,l)) -> +(x,sum(l))
    15: prod(nil()) -> |1|(|\#|())
    16: prod(cons(x,l)) -> *(x,prod(l))
Number of strict rules: 16
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #+(|1|(x),|0|(y)) -> #+(x,y)
   #2: #sum(nil()) -> #|0|(|\#|())
   #3: #*(|1|(x),y) -> #+(|0|(*(x,y)),y)
   #4: #*(|1|(x),y) -> #|0|(*(x,y))
   #5: #*(|1|(x),y) -> #*(x,y)
   #6: #*(*(x,y),z) -> #*(x,*(y,z))
   #7: #*(*(x,y),z) -> #*(y,z)
   #8: #sum(cons(x,l)) -> #+(x,sum(l))
   #9: #sum(cons(x,l)) -> #sum(l)
   #10: #+(|1|(x),|1|(y)) -> #|0|(+(+(x,y),|1|(|\#|())))
   #11: #+(|1|(x),|1|(y)) -> #+(+(x,y),|1|(|\#|()))
   #12: #+(|1|(x),|1|(y)) -> #+(x,y)
   #13: #*(|0|(x),y) -> #|0|(*(x,y))
   #14: #*(|0|(x),y) -> #*(x,y)
   #15: #+(|0|(x),|1|(y)) -> #+(x,y)
   #16: #prod(cons(x,l)) -> #*(x,prod(l))
   #17: #prod(cons(x,l)) -> #prod(l)
   #18: #+(+(x,y),z) -> #+(x,+(y,z))
   #19: #+(+(x,y),z) -> #+(y,z)
   #20: #+(|0|(x),|0|(y)) -> #|0|(+(x,y))
   #21: #+(|0|(x),|0|(y)) -> #+(x,y)
Number of SCCs: 4, DPs: 13, edges: 65
	SCC { #17 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|(x1)	weight: 0
  prod(x1)	weight: 0
 #|0|(x1)	weight: 0
#prod(x1)	weight: x1
   #*(x1,x2)	weight: 0
   sum(x1)	weight: 0
   nil()	weight: 0
   |1|(x1)	weight: 0
  cons(x1,x2)	weight: (/ 1 2) + x2
     +(x1,x2)	weight: 0
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
     *(x1,x2)	weight: 0
   |\#|()	weight: 0
    Usable rules: { }
    Removed DPs: #17
Number of SCCs: 3, DPs: 12, edges: 64
	SCC { #9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|(x1)	weight: 0
  prod(x1)	weight: 0
 #|0|(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: 0
   sum(x1)	weight: 0
   nil()	weight: 0
   |1|(x1)	weight: 0
  cons(x1,x2)	weight: (/ 1 2) + x2
     +(x1,x2)	weight: 0
 #sum(x1)	weight: x1
   #+(x1,x2)	weight: 0
     *(x1,x2)	weight: 0
   |\#|()	weight: 0
    Usable rules: { }
    Removed DPs: #9
Number of SCCs: 2, DPs: 11, edges: 63
	SCC { #5..7 #14 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|(x1)	weight: (/ 1 4) + x1
  prod(x1)	weight: 0
 #|0|(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: x1
   sum(x1)	weight: 0
   nil()	weight: 0
   |1|(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: (/ 1 4)
     +(x1,x2)	weight: (/ 3 4)
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
     *(x1,x2)	weight: (/ 1 4) + x1 + x2
   |\#|()	weight: 0
    Usable rules: { }
    Removed DPs: #5..7 #14
Number of SCCs: 1, DPs: 7, edges: 47
	SCC { #1 #11 #12 #15 #18 #19 #21 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|(x1)	weight: (/ 1 8) + x1
  prod(x1)	weight: 0
 #|0|(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: 0
   sum(x1)	weight: 0
   nil()	weight: 0
   |1|(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: (/ 1 8)
     +(x1,x2)	weight: (/ 1 8) + x1 + x2
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: x1 + x2
     *(x1,x2)	weight: x1
   |\#|()	weight: 0
    Usable rules: { 1..8 }
    Removed DPs: #1 #11 #12 #15 #19 #21
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #18 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|(x1)	weight: (/ 1 8) + x1
  prod(x1)	weight: 0
 #|0|(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: 0
   sum(x1)	weight: 0
   nil()	weight: 0
   |1|(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: (/ 1 8)
     +(x1,x2)	weight: (/ 1 8) + x1 + x2
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: x1
     *(x1,x2)	weight: x1
   |\#|()	weight: 0
    Usable rules: { 1..8 }
    Removed DPs: #18
Number of SCCs: 0, DPs: 0, edges: 0
YES