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 |2ndspos| |2ndsneg|
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: |2ndsneg|(|0|(),_1) ->= |2ndsneg|❆1_|0|(_1)
18: |2ndsneg|(s(_1),_2) ->= |2ndsneg|❆1_s(_1,_2)
19: |2ndspos|(|0|(),_1) ->= |2ndspos|❆1_|0|(_1)
20: |2ndspos|(s(_1),_2) ->= |2ndspos|❆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: #|2ndspos|(s(_1),_2) ->? #|2ndspos|❆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: #|2ndsneg|(|0|(),_1) ->? #|2ndsneg|❆1_|0|(_1)
   #11: #|2ndspos|(|0|(),_1) ->? #|2ndspos|❆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: #|2ndsneg|(s(_1),_2) ->? #|2ndsneg|❆1_s(_1,_2)
Number of SCCs: 3, DPs: 8, edges: 10
	SCC { #9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
negrecip(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
|2ndsneg|❆1_|0|(x1)	weight: 0
activate(x1)	weight: 0
  rnil()	weight: 0
|2ndspos|❆1_|0|(x1)	weight: 0
#|2ndsneg|❆1_|0|(x1)	weight: 0
|2ndspos|(x1,x2)	weight: 0
#plus(x1,x2)	weight: x1
n__from(x1)	weight: 0
square(x1)	weight: 0
#activate(x1)	weight: 0
#|2ndsneg|❆1_s(x1,x2)	weight: 0
#square(x1)	weight: 0
    pi(x1)	weight: 0
 rcons(x1,x2)	weight: 0
#|2ndspos|❆1_s(x1,x2)	weight: 0
|2ndsneg|(x1,x2)	weight: 0
#|2ndspos|(x1,x2)	weight: 0
#|2ndsneg|(x1,x2)	weight: 0
#times(x1,x2)	weight: 0
  from(x1)	weight: 0
 times(x1,x2)	weight: 0
|2ndspos|❆1_s(x1,x2)	weight: 0
#|2ndspos|❆1_|0|(x1)	weight: 0
  plus(x1,x2)	weight: 0
 cons2(x1,x2)	weight: 0
#from(x1)	weight: 0
  cons(x1,x2)	weight: 0
|2ndsneg|❆1_s(x1,x2)	weight: 0
  #pi(x1)	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.
   |0|()	weight: 0
negrecip(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
|2ndsneg|❆1_|0|(x1)	weight: 0
activate(x1)	weight: 0
  rnil()	weight: 0
|2ndspos|❆1_|0|(x1)	weight: 0
#|2ndsneg|❆1_|0|(x1)	weight: 0
|2ndspos|(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
n__from(x1)	weight: 0
square(x1)	weight: 0
#activate(x1)	weight: 0
#|2ndsneg|❆1_s(x1,x2)	weight: 0
#square(x1)	weight: 0
    pi(x1)	weight: 0
 rcons(x1,x2)	weight: 0
#|2ndspos|❆1_s(x1,x2)	weight: 0
|2ndsneg|(x1,x2)	weight: 0
#|2ndspos|(x1,x2)	weight: 0
#|2ndsneg|(x1,x2)	weight: 0
#times(x1,x2)	weight: x1
  from(x1)	weight: 0
 times(x1,x2)	weight: 0
|2ndspos|❆1_s(x1,x2)	weight: 0
#|2ndspos|❆1_|0|(x1)	weight: 0
  plus(x1,x2)	weight: 0
 cons2(x1,x2)	weight: 0
#from(x1)	weight: 0
  cons(x1,x2)	weight: 0
|2ndsneg|❆1_s(x1,x2)	weight: 0
  #pi(x1)	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.
   |0|()	weight: 0
negrecip(x1)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
|2ndsneg|❆1_|0|(x1)	weight: 0
activate(x1)	weight: (/ 1 4) + x1
  rnil()	weight: 0
|2ndspos|❆1_|0|(x1)	weight: 0
#|2ndsneg|❆1_|0|(x1)	weight: 0
|2ndspos|(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
n__from(x1)	weight: (/ 1 4) + x1
square(x1)	weight: 0
#activate(x1)	weight: 0
#|2ndsneg|❆1_s(x1,x2)	weight: (/ 1 4) + x1
#square(x1)	weight: 0
    pi(x1)	weight: 0
 rcons(x1,x2)	weight: 0
#|2ndspos|❆1_s(x1,x2)	weight: (/ 1 4) + x1
|2ndsneg|(x1,x2)	weight: 0
#|2ndspos|(x1,x2)	weight: x1
#|2ndsneg|(x1,x2)	weight: x1
#times(x1,x2)	weight: 0
  from(x1)	weight: (/ 3 4)
 times(x1,x2)	weight: 0
|2ndspos|❆1_s(x1,x2)	weight: 0
#|2ndspos|❆1_|0|(x1)	weight: 0
  plus(x1,x2)	weight: 0
 cons2(x1,x2)	weight: (/ 1 4)
#from(x1)	weight: 0
  cons(x1,x2)	weight: (/ 1 4) + x2
|2ndsneg|❆1_s(x1,x2)	weight: 0
  #pi(x1)	weight: 0
posrecip(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #6 #7 #17 #19
Number of SCCs: 0, DPs: 0, edges: 0
YES