Input TRS:
    1: a__from(X) -> cons(mark(X),from(s(X)))
    2: a__2ndspos(0(),Z) -> rnil()
    3: a__2ndspos(s(N),cons(X,Z)) -> a__2ndspos(s(mark(N)),cons2(X,mark(Z)))
    4: a__2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(mark(Y)),a__2ndsneg(mark(N),mark(Z)))
    5: a__2ndsneg(0(),Z) -> rnil()
    6: a__2ndsneg(s(N),cons(X,Z)) -> a__2ndsneg(s(mark(N)),cons2(X,mark(Z)))
    7: a__2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(mark(Y)),a__2ndspos(mark(N),mark(Z)))
    8: a__pi(X) -> a__2ndspos(mark(X),a__from(0()))
    9: a__plus(0(),Y) -> mark(Y)
    10: a__plus(s(X),Y) -> s(a__plus(mark(X),mark(Y)))
    11: a__times(0(),Y) -> 0()
    12: a__times(s(X),Y) -> a__plus(mark(Y),a__times(mark(X),mark(Y)))
    13: a__square(X) -> a__times(mark(X),mark(X))
    14: mark(from(X)) -> a__from(mark(X))
    15: mark(2ndspos(X1,X2)) -> a__2ndspos(mark(X1),mark(X2))
    16: mark(2ndsneg(X1,X2)) -> a__2ndsneg(mark(X1),mark(X2))
    17: mark(pi(X)) -> a__pi(mark(X))
    18: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
    19: mark(times(X1,X2)) -> a__times(mark(X1),mark(X2))
    20: mark(square(X)) -> a__square(mark(X))
    21: mark(0()) -> 0()
    22: mark(s(X)) -> s(mark(X))
    23: mark(posrecip(X)) -> posrecip(mark(X))
    24: mark(negrecip(X)) -> negrecip(mark(X))
    25: mark(nil()) -> nil()
    26: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    27: mark(cons2(X1,X2)) -> cons2(X1,mark(X2))
    28: mark(rnil()) -> rnil()
    29: mark(rcons(X1,X2)) -> rcons(mark(X1),mark(X2))
    30: a__from(X) -> from(X)
    31: a__2ndspos(X1,X2) -> 2ndspos(X1,X2)
    32: a__2ndsneg(X1,X2) -> 2ndsneg(X1,X2)
    33: a__pi(X) -> pi(X)
    34: a__plus(X1,X2) -> plus(X1,X2)
    35: a__times(X1,X2) -> times(X1,X2)
    36: a__square(X) -> square(X)
Number of strict rules: 36
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #mark(rcons(X1,X2)) -> #mark(X1)
   #2: #mark(rcons(X1,X2)) -> #mark(X2)
   #3: #a__2ndsneg(s(N),cons(X,Z)) -> #a__2ndsneg(s(mark(N)),cons2(X,mark(Z)))
   #4: #a__2ndsneg(s(N),cons(X,Z)) -> #mark(N)
   #5: #a__2ndsneg(s(N),cons(X,Z)) -> #mark(Z)
   #6: #a__square(X) -> #a__times(mark(X),mark(X))
   #7: #a__square(X) -> #mark(X)
   #8: #a__square(X) -> #mark(X)
   #9: #a__plus(0(),Y) -> #mark(Y)
   #10: #mark(negrecip(X)) -> #mark(X)
   #11: #mark(posrecip(X)) -> #mark(X)
   #12: #a__times(s(X),Y) -> #a__plus(mark(Y),a__times(mark(X),mark(Y)))
   #13: #a__times(s(X),Y) -> #mark(Y)
   #14: #a__times(s(X),Y) -> #a__times(mark(X),mark(Y))
   #15: #a__times(s(X),Y) -> #mark(X)
   #16: #a__times(s(X),Y) -> #mark(Y)
   #17: #mark(from(X)) -> #a__from(mark(X))
   #18: #mark(from(X)) -> #mark(X)
   #19: #mark(square(X)) -> #a__square(mark(X))
   #20: #mark(square(X)) -> #mark(X)
   #21: #a__2ndsneg(s(N),cons2(X,cons(Y,Z))) -> #mark(Y)
   #22: #a__2ndsneg(s(N),cons2(X,cons(Y,Z))) -> #a__2ndspos(mark(N),mark(Z))
   #23: #a__2ndsneg(s(N),cons2(X,cons(Y,Z))) -> #mark(N)
   #24: #a__2ndsneg(s(N),cons2(X,cons(Y,Z))) -> #mark(Z)
   #25: #a__plus(s(X),Y) -> #a__plus(mark(X),mark(Y))
   #26: #a__plus(s(X),Y) -> #mark(X)
   #27: #a__plus(s(X),Y) -> #mark(Y)
   #28: #mark(s(X)) -> #mark(X)
   #29: #mark(cons2(X1,X2)) -> #mark(X2)
   #30: #mark(pi(X)) -> #a__pi(mark(X))
   #31: #mark(pi(X)) -> #mark(X)
   #32: #mark(times(X1,X2)) -> #a__times(mark(X1),mark(X2))
   #33: #mark(times(X1,X2)) -> #mark(X1)
   #34: #mark(times(X1,X2)) -> #mark(X2)
   #35: #mark(cons(X1,X2)) -> #mark(X1)
   #36: #mark(2ndsneg(X1,X2)) -> #a__2ndsneg(mark(X1),mark(X2))
   #37: #mark(2ndsneg(X1,X2)) -> #mark(X1)
   #38: #mark(2ndsneg(X1,X2)) -> #mark(X2)
   #39: #a__2ndspos(s(N),cons(X,Z)) -> #a__2ndspos(s(mark(N)),cons2(X,mark(Z)))
   #40: #a__2ndspos(s(N),cons(X,Z)) -> #mark(N)
   #41: #a__2ndspos(s(N),cons(X,Z)) -> #mark(Z)
   #42: #a__from(X) -> #mark(X)
   #43: #a__pi(X) -> #a__2ndspos(mark(X),a__from(0()))
   #44: #a__pi(X) -> #mark(X)
   #45: #a__pi(X) -> #a__from(0())
   #46: #mark(2ndspos(X1,X2)) -> #a__2ndspos(mark(X1),mark(X2))
   #47: #mark(2ndspos(X1,X2)) -> #mark(X1)
   #48: #mark(2ndspos(X1,X2)) -> #mark(X2)
   #49: #a__2ndspos(s(N),cons2(X,cons(Y,Z))) -> #mark(Y)
   #50: #a__2ndspos(s(N),cons2(X,cons(Y,Z))) -> #a__2ndsneg(mark(N),mark(Z))
   #51: #a__2ndspos(s(N),cons2(X,cons(Y,Z))) -> #mark(N)
   #52: #a__2ndspos(s(N),cons2(X,cons(Y,Z))) -> #mark(Z)
   #53: #mark(plus(X1,X2)) -> #a__plus(mark(X1),mark(X2))
   #54: #mark(plus(X1,X2)) -> #mark(X1)
   #55: #mark(plus(X1,X2)) -> #mark(X2)
Number of SCCs: 1, DPs: 55, edges: 1024
	SCC { #1..55 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
a__plus(x1,x2)	weight: max{x2, (/ 1 8) + x1}
a__2ndsneg(x1,x2)	weight: max{(/ 1 2) + x2, (/ 3 8) + x1}
negrecip(x1)	weight: (/ 3 4) + x1
     s(x1)	weight: x1
#a__pi(x1)	weight: (/ 103629 4) + x1
#a__from(x1)	weight: (/ 1 4) + x1
2ndspos(x1,x2)	weight: max{(/ 1 2) + x2, (/ 3 8) + x1}
a__from(x1)	weight: (/ 3 8) + x1
  rnil()	weight: (/ 5 8)
square(x1)	weight: (/ 207253 8) + x1
#a__times(x1,x2)	weight: max{(/ 207251 8) + x2, (/ 51813 2) + x1}
    pi(x1)	weight: (/ 103629 4) + x1
 rcons(x1,x2)	weight: max{x2, (/ 1 8) + x1}
a__2ndspos(x1,x2)	weight: max{(/ 1 2) + x2, (/ 3 8) + x1}
#a__2ndsneg(x1,x2)	weight: max{(/ 1 4) + x2, (/ 3 8) + x1}
#a__plus(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 4) + x1}
#mark(x1)	weight: (/ 1 8) + x1
     0()	weight: (/ 207251 8)
  from(x1)	weight: (/ 3 8) + x1
 times(x1,x2)	weight: max{(/ 103625 4) + x2, (/ 207251 8) + x1}
 a__pi(x1)	weight: (/ 103629 4) + x1
   nil()	weight: (/ 1 8)
  mark(x1)	weight: x1
2ndsneg(x1,x2)	weight: max{(/ 1 2) + x2, (/ 3 8) + x1}
  plus(x1,x2)	weight: max{x2, (/ 1 8) + x1}
 cons2(x1,x2)	weight: max{x2, (/ 1 4) + x1}
a__square(x1)	weight: (/ 207253 8) + x1
  cons(x1,x2)	weight: max{x2, (/ 3 8) + x1}
a__times(x1,x2)	weight: max{(/ 103625 4) + x2, (/ 207251 8) + x1}
#a__square(x1)	weight: (/ 207253 8) + x1
#a__2ndspos(x1,x2)	weight: max{(/ 1 4) + x2, (/ 3 8) + x1}
posrecip(x1)	weight: (/ 3 4) + x1
    Usable rules: { 1..36 }
    Removed DPs: #1 #4..8 #10 #11 #13 #15..21 #23 #24 #26 #30 #31 #33..38 #40..49 #51 #52 #54
Number of SCCs: 2, DPs: 15, edges: 55
	SCC { #3 #22 #39 #50 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
a__plus(x1,x2)	status: [x2,x1]	precedence above: s plus
a__2ndsneg(x1,x2)	status: []	precedence above: rnil rcons 2ndsneg
negrecip(x1)	status: x1
     s(x1)	status: [x1]	precedence above:
#a__pi(x1)	status: []	precedence above:
#a__from(x1)	status: []	precedence above:
2ndspos(x1,x2)	status: [x2]	precedence above: a__2ndsneg s rnil rcons a__2ndspos mark 2ndsneg cons2 posrecip
a__from(x1)	status: [x1]	precedence above: a__2ndsneg s rnil rcons from mark 2ndsneg cons2 cons posrecip
  rnil()	status: []	precedence above:
square(x1)	status: [x1]	precedence above: a__plus s times mark plus a__square a__times
#a__times(x1,x2)	status: [x1]	precedence above:
    pi(x1)	status: [x1]	precedence above: a__2ndsneg s 2ndspos a__from rnil rcons a__2ndspos 0 from a__pi mark 2ndsneg cons2 cons posrecip
 rcons(x1,x2)	status: []	precedence above:
a__2ndspos(x1,x2)	status: [x2]	precedence above: a__2ndsneg s 2ndspos rnil rcons mark 2ndsneg cons2 posrecip
#a__2ndsneg(x1,x2)	status: x1
#a__plus(x1,x2)	status: [x2,x1]	precedence above:
#mark(x1)	status: []	precedence above:
     0()	status: []	precedence above: rnil
  from(x1)	status: [x1]	precedence above: a__2ndsneg s a__from rnil rcons mark 2ndsneg cons2 cons posrecip
 times(x1,x2)	status: [x1,x2]	precedence above: a__plus s mark plus a__times
 a__pi(x1)	status: [x1]	precedence above: a__2ndsneg s 2ndspos a__from rnil pi rcons a__2ndspos 0 from mark 2ndsneg cons2 cons posrecip
   nil()	status: []	precedence above:
  mark(x1)	status: x1
2ndsneg(x1,x2)	status: []	precedence above: a__2ndsneg rnil rcons
  plus(x1,x2)	status: [x2,x1]	precedence above: a__plus s
 cons2(x1,x2)	status: [x1]	precedence above: s mark
a__square(x1)	status: [x1]	precedence above: a__plus s square times mark plus a__times
  cons(x1,x2)	status: [x1]	precedence above: a__2ndsneg s rnil rcons mark 2ndsneg cons2 posrecip
a__times(x1,x2)	status: [x1,x2]	precedence above: a__plus s times mark plus
#a__square(x1)	status: []	precedence above:
#a__2ndspos(x1,x2)	status: x1
posrecip(x1)	status: [x1]	precedence above: s mark
    Usable rules: { 1..36 }
    Removed DPs: #22 #50
Number of SCCs: 1, DPs: 11, edges: 49
	SCC { #2 #9 #12 #14 #25 #27..29 #32 #53 #55 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
a__plus(x1,x2)	weight: x2
a__2ndsneg(x1,x2)	weight: (/ 1 4)
negrecip(x1)	weight: (/ 1 4)
     s(x1)	weight: x1
#a__pi(x1)	weight: 0
#a__from(x1)	weight: 0
2ndspos(x1,x2)	weight: (/ 1 4)
a__from(x1)	weight: (/ 1 4)
  rnil()	weight: 0
square(x1)	weight: (/ 1 4) + x1
#a__times(x1,x2)	weight: (/ 1 4)
    pi(x1)	weight: (/ 1 4)
 rcons(x1,x2)	weight: x2
a__2ndspos(x1,x2)	weight: (/ 1 4)
#a__2ndsneg(x1,x2)	weight: 0
#a__plus(x1,x2)	weight: x2
#mark(x1)	weight: x1
     0()	weight: 0
  from(x1)	weight: (/ 1 4)
 times(x1,x2)	weight: (/ 1 4)
 a__pi(x1)	weight: (/ 1 4)
   nil()	weight: 0
  mark(x1)	weight: x1
2ndsneg(x1,x2)	weight: (/ 1 4)
  plus(x1,x2)	weight: x2
 cons2(x1,x2)	weight: (/ 1 4) + x1 + x2
a__square(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: (/ 1 4)
a__times(x1,x2)	weight: (/ 1 4)
#a__square(x1)	weight: 0
#a__2ndspos(x1,x2)	weight: (/ 1 4)
posrecip(x1)	weight: (/ 1 4)
    Usable rules: { 1..36 }
    Removed DPs: #29
Number of SCCs: 1, DPs: 10, edges: 38
	SCC { #2 #9 #12 #14 #25 #27 #28 #32 #53 #55 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
a__plus(x1,x2)	status: [x2,x1]	precedence above: s #a__plus plus
a__2ndsneg(x1,x2)	status: [x1]	precedence above: 2ndspos rnil rcons a__2ndspos 2ndsneg
negrecip(x1)	status: x1
     s(x1)	status: [x1]	precedence above:
#a__pi(x1)	status: []	precedence above:
#a__from(x1)	status: []	precedence above:
2ndspos(x1,x2)	status: [x1]	precedence above: a__2ndsneg rnil rcons a__2ndspos 2ndsneg
a__from(x1)	status: [x1]	precedence above: a__2ndsneg s 2ndspos rnil rcons a__2ndspos from mark 2ndsneg cons2 cons posrecip
  rnil()	status: []	precedence above:
square(x1)	status: [x1]	precedence above: a__plus s #a__times #a__plus times mark plus a__square a__times
#a__times(x1,x2)	status: [x1,x2]	precedence above: a__plus s #a__plus times mark plus a__times
    pi(x1)	status: [x1]	precedence above: a__2ndsneg s 2ndspos a__from rnil rcons a__2ndspos 0 from a__pi mark 2ndsneg cons2 cons posrecip
 rcons(x1,x2)	status: [x2]	precedence above:
a__2ndspos(x1,x2)	status: [x1]	precedence above: a__2ndsneg 2ndspos rnil rcons 2ndsneg
#a__2ndsneg(x1,x2)	status: x1
#a__plus(x1,x2)	status: [x2,x1]	precedence above: a__plus s plus
#mark(x1)	status: x1
     0()	status: []	precedence above: rnil
  from(x1)	status: [x1]	precedence above: a__2ndsneg s 2ndspos a__from rnil rcons a__2ndspos mark 2ndsneg cons2 cons posrecip
 times(x1,x2)	status: [x1,x2]	precedence above: a__plus s #a__times #a__plus mark plus a__times
 a__pi(x1)	status: [x1]	precedence above: a__2ndsneg s 2ndspos a__from rnil pi rcons a__2ndspos 0 from mark 2ndsneg cons2 cons posrecip
   nil()	status: []	precedence above:
  mark(x1)	status: x1
2ndsneg(x1,x2)	status: [x1]	precedence above: a__2ndsneg 2ndspos rnil rcons a__2ndspos
  plus(x1,x2)	status: [x2,x1]	precedence above: a__plus s #a__plus
 cons2(x1,x2)	status: [x1]	precedence above: s mark
a__square(x1)	status: [x1]	precedence above: a__plus s square #a__times #a__plus times mark plus a__times
  cons(x1,x2)	status: [x1]	precedence above: a__2ndsneg s 2ndspos rnil rcons a__2ndspos mark 2ndsneg cons2 posrecip
a__times(x1,x2)	status: [x1,x2]	precedence above: a__plus s #a__times #a__plus times mark plus
#a__square(x1)	status: []	precedence above:
#a__2ndspos(x1,x2)	status: x1
posrecip(x1)	status: [x1]	precedence above: s mark
    Usable rules: { 1..36 }
    Removed DPs: #2 #9 #12 #14 #25 #27 #28 #55
Number of SCCs: 0, DPs: 0, edges: 0
YES