Input TRS:
    1: from(X) -> cons(X,n__from(s(X)))
    2: 2ndspos(0(),Z) -> rnil()
    3: 2ndspos(s(N),cons(X,Z)) -> 2ndspos(s(N),cons2(X,activate(Z)))
    4: 2ndspos(s(N),cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
    5: 2ndsneg(0(),Z) -> rnil()
    6: 2ndsneg(s(N),cons(X,Z)) -> 2ndsneg(s(N),cons2(X,activate(Z)))
    7: 2ndsneg(s(N),cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
    8: pi(X) -> 2ndspos(X,from(0()))
    9: plus(0(),Y) -> Y
    10: plus(s(X),Y) -> s(plus(X,Y))
    11: times(0(),Y) -> 0()
    12: times(s(X),Y) -> plus(Y,times(X,Y))
    13: square(X) -> times(X,X)
    14: from(X) -> n__from(X)
    15: activate(n__from(X)) -> from(X)
    16: activate(X) -> X
Number of strict rules: 16
Direct Order(PosReal,>,Poly) ... failed.
Freezing 2ndsneg 2ndspos
1: from(X) -> cons(X,n__from(s(X)))
2: 2ndspos❆1_0(Z) -> rnil()
3: 2ndspos❆1_s(N,cons(X,Z)) -> 2ndspos❆1_s(N,cons2(X,activate(Z)))
4: 2ndspos❆1_s(N,cons2(X,cons(Y,Z))) -> rcons(posrecip(Y),2ndsneg(N,activate(Z)))
5: 2ndsneg❆1_0(Z) -> rnil()
6: 2ndsneg❆1_s(N,cons(X,Z)) -> 2ndsneg❆1_s(N,cons2(X,activate(Z)))
7: 2ndsneg❆1_s(N,cons2(X,cons(Y,Z))) -> rcons(negrecip(Y),2ndspos(N,activate(Z)))
8: pi(X) -> 2ndspos(X,from(0()))
9: plus(0(),Y) -> Y
10: plus(s(X),Y) -> s(plus(X,Y))
11: times(0(),Y) -> 0()
12: times(s(X),Y) -> plus(Y,times(X,Y))
13: square(X) -> times(X,X)
14: from(X) -> n__from(X)
15: activate(n__from(X)) -> from(X)
16: activate(X) -> X
17: 2ndspos(0(),_1) ->= 2ndspos❆1_0(_1)
18: 2ndspos(s(_1),_2) ->= 2ndspos❆1_s(_1,_2)
19: 2ndsneg(0(),_1) ->= 2ndsneg❆1_0(_1)
20: 2ndsneg(s(_1),_2) ->= 2ndsneg❆1_s(_1,_2)
Number of strict rules: 16
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #2ndsneg❆1_s(N,cons(X,Z)) -> #2ndsneg❆1_s(N,cons2(X,activate(Z)))
   #2: #2ndsneg❆1_s(N,cons(X,Z)) -> #activate(Z)
   #3: #square(X) -> #times(X,X)
   #4: #times(s(X),Y) -> #plus(Y,times(X,Y))
   #5: #times(s(X),Y) -> #times(X,Y)
   #6: #2ndsneg(s(_1),_2) ->? #2ndsneg❆1_s(_1,_2)
   #7: #2ndsneg❆1_s(N,cons2(X,cons(Y,Z))) -> #2ndspos(N,activate(Z))
   #8: #2ndsneg❆1_s(N,cons2(X,cons(Y,Z))) -> #activate(Z)
   #9: #plus(s(X),Y) -> #plus(X,Y)
   #10: #2ndspos(0(),_1) ->? #2ndspos❆1_0(_1)
   #11: #2ndsneg(0(),_1) ->? #2ndsneg❆1_0(_1)
   #12: #2ndspos❆1_s(N,cons(X,Z)) -> #2ndspos❆1_s(N,cons2(X,activate(Z)))
   #13: #2ndspos❆1_s(N,cons(X,Z)) -> #activate(Z)
   #14: #pi(X) -> #2ndspos(X,from(0()))
   #15: #pi(X) -> #from(0())
   #16: #activate(n__from(X)) -> #from(X)
   #17: #2ndspos❆1_s(N,cons2(X,cons(Y,Z))) -> #2ndsneg(N,activate(Z))
   #18: #2ndspos❆1_s(N,cons2(X,cons(Y,Z))) -> #activate(Z)
   #19: #2ndspos(s(_1),_2) ->? #2ndspos❆1_s(_1,_2)
Number of SCCs: 3, DPs: 8, edges: 10
	SCC { #9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
2ndspos❆1_s(x1,x2)	weight: 0
negrecip(x1)	weight: 0
#2ndspos❆1_s(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#2ndsneg❆1_0(x1)	weight: 0
#2ndspos❆1_0(x1)	weight: 0
2ndspos(x1,x2)	weight: 0
2ndsneg❆1_0(x1)	weight: 0
2ndsneg❆1_s(x1,x2)	weight: 0
activate(x1)	weight: 0
2ndspos❆1_0(x1)	weight: 0
  rnil()	weight: 0
#plus(x1,x2)	weight: x1
#2ndsneg❆1_s(x1,x2)	weight: 0
n__from(x1)	weight: 0
square(x1)	weight: 0
#activate(x1)	weight: 0
#square(x1)	weight: 0
    pi(x1)	weight: 0
 rcons(x1,x2)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
  from(x1)	weight: 0
 times(x1,x2)	weight: 0
2ndsneg(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
#2ndspos(x1,x2)	weight: 0
 cons2(x1,x2)	weight: 0
#from(x1)	weight: 0
  cons(x1,x2)	weight: 0
  #pi(x1)	weight: 0
#2ndsneg(x1,x2)	weight: 0
posrecip(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #9
Number of SCCs: 2, DPs: 7, edges: 9
	SCC { #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
2ndspos❆1_s(x1,x2)	weight: 0
negrecip(x1)	weight: 0
#2ndspos❆1_s(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#2ndsneg❆1_0(x1)	weight: 0
#2ndspos❆1_0(x1)	weight: 0
2ndspos(x1,x2)	weight: 0
2ndsneg❆1_0(x1)	weight: 0
2ndsneg❆1_s(x1,x2)	weight: 0
activate(x1)	weight: 0
2ndspos❆1_0(x1)	weight: 0
  rnil()	weight: 0
#plus(x1,x2)	weight: 0
#2ndsneg❆1_s(x1,x2)	weight: 0
n__from(x1)	weight: 0
square(x1)	weight: 0
#activate(x1)	weight: 0
#square(x1)	weight: 0
    pi(x1)	weight: 0
 rcons(x1,x2)	weight: 0
#times(x1,x2)	weight: x1
     0()	weight: 0
  from(x1)	weight: 0
 times(x1,x2)	weight: 0
2ndsneg(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
#2ndspos(x1,x2)	weight: 0
 cons2(x1,x2)	weight: 0
#from(x1)	weight: 0
  cons(x1,x2)	weight: 0
  #pi(x1)	weight: 0
#2ndsneg(x1,x2)	weight: 0
posrecip(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #5
Number of SCCs: 1, DPs: 6, edges: 8
	SCC { #1 #6 #7 #12 #17 #19 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
2ndspos❆1_s(x1,x2)	weight: 0
negrecip(x1)	weight: 0
#2ndspos❆1_s(x1,x2)	weight: (/ 1 4) + x1
     s(x1)	weight: (/ 1 2) + x1
#2ndsneg❆1_0(x1)	weight: 0
#2ndspos❆1_0(x1)	weight: 0
2ndspos(x1,x2)	weight: 0
2ndsneg❆1_0(x1)	weight: 0
2ndsneg❆1_s(x1,x2)	weight: 0
activate(x1)	weight: (/ 1 4)
2ndspos❆1_0(x1)	weight: 0
  rnil()	weight: 0
#plus(x1,x2)	weight: 0
#2ndsneg❆1_s(x1,x2)	weight: (/ 1 4) + x1
n__from(x1)	weight: (/ 3 4)
square(x1)	weight: 0
#activate(x1)	weight: 0
#square(x1)	weight: 0
    pi(x1)	weight: 0
 rcons(x1,x2)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
  from(x1)	weight: (/ 1 2) + x1
 times(x1,x2)	weight: 0
2ndsneg(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
#2ndspos(x1,x2)	weight: x1
 cons2(x1,x2)	weight: (/ 1 4)
#from(x1)	weight: 0
  cons(x1,x2)	weight: (/ 3 4)
  #pi(x1)	weight: 0
#2ndsneg(x1,x2)	weight: x1
posrecip(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #6 #7 #17 #19
Number of SCCs: 0, DPs: 0, edges: 0
YES