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: *(x,+(y,z)) -> +(*(x,y),*(x,z))
    14: app(nil(),l) -> l
    15: app(cons(x,l1),l2) -> cons(x,app(l1,l2))
    16: sum(nil()) -> |0|(|\#|())
    17: sum(cons(x,l)) -> +(x,sum(l))
    18: sum(app(l1,l2)) -> +(sum(l1),sum(l2))
    19: prod(nil()) -> |1|(|\#|())
    20: prod(cons(x,l)) -> *(x,prod(l))
    21: prod(app(l1,l2)) -> *(prod(l1),prod(l2))
Number of strict rules: 21
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #+(|1|(x),|0|(y)) -> #+(x,y)
   #2: #*(x,+(y,z)) -> #+(*(x,y),*(x,z))
   #3: #*(x,+(y,z)) -> #*(x,y)
   #4: #*(x,+(y,z)) -> #*(x,z)
   #5: #*(|1|(x),y) -> #+(|0|(*(x,y)),y)
   #6: #*(|1|(x),y) -> #|0|(*(x,y))
   #7: #*(|1|(x),y) -> #*(x,y)
   #8: #*(*(x,y),z) -> #*(x,*(y,z))
   #9: #*(*(x,y),z) -> #*(y,z)
   #10: #prod(cons(x,l)) -> #*(x,prod(l))
   #11: #prod(cons(x,l)) -> #prod(l)
   #12: #+(|1|(x),|1|(y)) -> #|0|(+(+(x,y),|1|(|\#|())))
   #13: #+(|1|(x),|1|(y)) -> #+(+(x,y),|1|(|\#|()))
   #14: #+(|1|(x),|1|(y)) -> #+(x,y)
   #15: #*(|0|(x),y) -> #|0|(*(x,y))
   #16: #*(|0|(x),y) -> #*(x,y)
   #17: #+(|0|(x),|1|(y)) -> #+(x,y)
   #18: #sum(cons(x,l)) -> #+(x,sum(l))
   #19: #sum(cons(x,l)) -> #sum(l)
   #20: #prod(app(l1,l2)) -> #*(prod(l1),prod(l2))
   #21: #prod(app(l1,l2)) -> #prod(l1)
   #22: #prod(app(l1,l2)) -> #prod(l2)
   #23: #sum(nil()) -> #|0|(|\#|())
   #24: #+(+(x,y),z) -> #+(x,+(y,z))
   #25: #+(+(x,y),z) -> #+(y,z)
   #26: #app(cons(x,l1),l2) -> #app(l1,l2)
   #27: #+(|0|(x),|0|(y)) -> #|0|(+(x,y))
   #28: #+(|0|(x),|0|(y)) -> #+(x,y)
   #29: #sum(app(l1,l2)) -> #+(sum(l1),sum(l2))
   #30: #sum(app(l1,l2)) -> #sum(l1)
   #31: #sum(app(l1,l2)) -> #sum(l2)
Number of SCCs: 5, DPs: 20, edges: 102
	SCC { #26 }
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
 #app(x1,x2)	weight: x1
  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
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #26
Number of SCCs: 4, DPs: 19, edges: 101
	SCC { #11 #21 #22 }
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
 #app(x1,x2)	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
   app(x1,x2)	weight: (/ 1 2) + x1 + x2
    Usable rules: { }
    Removed DPs: #11 #21 #22
Number of SCCs: 3, DPs: 16, edges: 92
	SCC { #19 #30 #31 }
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
 #app(x1,x2)	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
   app(x1,x2)	weight: (/ 1 2) + x1 + x2
    Usable rules: { }
    Removed DPs: #19 #30 #31
Number of SCCs: 2, DPs: 13, edges: 83
	SCC { #3 #4 #7..9 #16 }
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
 #app(x1,x2)	weight: 0
  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
   app(x1,x2)	weight: (/ 1 4)
    Usable rules: { }
    Removed DPs: #7..9 #16
Number of SCCs: 2, DPs: 9, edges: 51
	SCC { #3 #4 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
   |0|(x1)	weight: 0
  prod(x1)	weight: 0
 #|0|(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: max{0, (/ 1 4) + x2}
   sum(x1)	weight: 0
   nil()	weight: 0
   |1|(x1)	weight: 0
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: 0
     +(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: 0
     *(x1,x2)	weight: 0
   |\#|()	weight: 0
   app(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #3 #4
Number of SCCs: 1, DPs: 7, edges: 47
	SCC { #1 #13 #14 #17 #24 #25 #28 }
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
 #app(x1,x2)	weight: 0
  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: (/ 1 8) + x1
   |\#|()	weight: 0
   app(x1,x2)	weight: (/ 1 8)
    Usable rules: { 1..8 }
    Removed DPs: #1 #13 #14 #17 #25 #28
Number of SCCs: 2, DPs: 1, edges: 1
	SCC { #24 }
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
 #app(x1,x2)	weight: 0
  cons(x1,x2)	weight: (/ 1 8)
     +(x1,x2)	weight: (/ 1 8) + x1 + x2
 #sum(x1)	weight: 0
   #+(x1,x2)	weight: x1
     *(x1,x2)	weight: (/ 1 8) + x1
   |\#|()	weight: 0
   app(x1,x2)	weight: (/ 1 8)
    Usable rules: { 1..8 }
    Removed DPs: #24
Number of SCCs: 1, DPs: 0, edges: 0
YES