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
     1(x1)	weight: 0
  prod(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: 0
     \#()	weight: 0
   sum(x1)	weight: 0
     0(x1)	weight: 0
   nil()	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
   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
     1(x1)	weight: 0
  prod(x1)	weight: 0
#prod(x1)	weight: x1
   #*(x1,x2)	weight: 0
     \#()	weight: 0
   sum(x1)	weight: 0
     0(x1)	weight: 0
   nil()	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
   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
     1(x1)	weight: 0
  prod(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: 0
     \#()	weight: 0
   sum(x1)	weight: 0
     0(x1)	weight: 0
   nil()	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
   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: 0
     1(x1)	weight: (/ 1 4) + x1
  prod(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: x1
     \#()	weight: 0
   sum(x1)	weight: 0
     0(x1)	weight: (/ 1 4) + x1
   nil()	weight: 0
 #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
   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
     1(x1)	weight: 0
  prod(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: max{0, (/ 1 4) + x2}
     \#()	weight: 0
   sum(x1)	weight: 0
     0(x1)	weight: 0
   nil()	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
   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: 0
     1(x1)	weight: (/ 1 4) + x1
  prod(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: 0
     \#()	weight: 0
   sum(x1)	weight: 0
     0(x1)	weight: (/ 1 8) + x1
   nil()	weight: 0
 #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 + x2
   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: 0
     1(x1)	weight: (/ 1 4) + x1
  prod(x1)	weight: 0
#prod(x1)	weight: 0
   #*(x1,x2)	weight: 0
     \#()	weight: 0
   sum(x1)	weight: 0
     0(x1)	weight: (/ 1 8) + x1
   nil()	weight: 0
 #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 + x2
   app(x1,x2)	weight: (/ 1 8)
    Usable rules: { 1..8 }
    Removed DPs: #24
Number of SCCs: 1, DPs: 0, edges: 0
YES