Input TRS:
    1: terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
    2: sqr(0()) -> 0()
    3: sqr(s(X)) -> s(add(sqr(X),dbl(X)))
    4: dbl(0()) -> 0()
    5: dbl(s(X)) -> s(s(dbl(X)))
    6: add(0(),X) -> X
    7: add(s(X),Y) -> s(add(X,Y))
    8: first(0(),X) -> nil()
    9: first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
    10: terms(X) -> n__terms(X)
    11: s(X) -> n__s(X)
    12: first(X1,X2) -> n__first(X1,X2)
    13: activate(n__terms(X)) -> terms(activate(X))
    14: activate(n__s(X)) -> s(activate(X))
    15: activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
    16: activate(X) -> X
Number of strict rules: 16
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #activate(n__terms(X)) -> #terms(activate(X))
   #2: #activate(n__terms(X)) -> #activate(X)
   #3: #first(s(X),cons(Y,Z)) -> #activate(Z)
   #4: #activate(n__s(X)) -> #s(activate(X))
   #5: #activate(n__s(X)) -> #activate(X)
   #6: #add(s(X),Y) -> #s(add(X,Y))
   #7: #add(s(X),Y) -> #add(X,Y)
   #8: #dbl(s(X)) -> #s(s(dbl(X)))
   #9: #dbl(s(X)) -> #s(dbl(X))
   #10: #dbl(s(X)) -> #dbl(X)
   #11: #sqr(s(X)) -> #s(add(sqr(X),dbl(X)))
   #12: #sqr(s(X)) -> #add(sqr(X),dbl(X))
   #13: #sqr(s(X)) -> #sqr(X)
   #14: #sqr(s(X)) -> #dbl(X)
   #15: #terms(N) -> #sqr(N)
   #16: #activate(n__first(X1,X2)) -> #first(activate(X1),activate(X2))
   #17: #activate(n__first(X1,X2)) -> #activate(X1)
   #18: #activate(n__first(X1,X2)) -> #activate(X2)
Number of SCCs: 4, DPs: 9, edges: 29
	SCC { #10 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
n__first(x1,x2)	weight: 0
 recip(x1)	weight: 0
activate(x1)	weight: 0
   dbl(x1)	weight: 0
 #dbl(x1)	weight: x1
#terms(x1)	weight: 0
#activate(x1)	weight: 0
  n__s(x1)	weight: 0
 #sqr(x1)	weight: 0
     0()	weight: 0
   #s(x1)	weight: 0
#first(x1,x2)	weight: 0
   nil()	weight: 0
 first(x1,x2)	weight: 0
n__terms(x1)	weight: 0
  cons(x1,x2)	weight: 0
 #add(x1,x2)	weight: 0
   add(x1,x2)	weight: 0
   sqr(x1)	weight: 0
 terms(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #10
Number of SCCs: 3, DPs: 8, edges: 28
	SCC { #13 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
n__first(x1,x2)	weight: 0
 recip(x1)	weight: 0
activate(x1)	weight: 0
   dbl(x1)	weight: 0
 #dbl(x1)	weight: 0
#terms(x1)	weight: 0
#activate(x1)	weight: 0
  n__s(x1)	weight: 0
 #sqr(x1)	weight: x1
     0()	weight: 0
   #s(x1)	weight: 0
#first(x1,x2)	weight: 0
   nil()	weight: 0
 first(x1,x2)	weight: 0
n__terms(x1)	weight: 0
  cons(x1,x2)	weight: 0
 #add(x1,x2)	weight: 0
   add(x1,x2)	weight: 0
   sqr(x1)	weight: 0
 terms(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #13
Number of SCCs: 2, DPs: 7, edges: 27
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
n__first(x1,x2)	weight: 0
 recip(x1)	weight: 0
activate(x1)	weight: 0
   dbl(x1)	weight: 0
 #dbl(x1)	weight: 0
#terms(x1)	weight: 0
#activate(x1)	weight: 0
  n__s(x1)	weight: 0
 #sqr(x1)	weight: 0
     0()	weight: 0
   #s(x1)	weight: 0
#first(x1,x2)	weight: 0
   nil()	weight: 0
 first(x1,x2)	weight: 0
n__terms(x1)	weight: 0
  cons(x1,x2)	weight: 0
 #add(x1,x2)	weight: x1
   add(x1,x2)	weight: 0
   sqr(x1)	weight: 0
 terms(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #7
Number of SCCs: 1, DPs: 6, edges: 26
	SCC { #2 #3 #5 #16..18 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: x1
n__first(x1,x2)	weight: (/ 1 2) + x1 + x2
 recip(x1)	weight: (/ 1 4)
activate(x1)	weight: x1
   dbl(x1)	weight: (/ 1 4)
 #dbl(x1)	weight: 0
#terms(x1)	weight: 0
#activate(x1)	weight: x1
  n__s(x1)	weight: x1
 #sqr(x1)	weight: 0
     0()	weight: 0
   #s(x1)	weight: 0
#first(x1,x2)	weight: (/ 1 4) + x2
   nil()	weight: 0
 first(x1,x2)	weight: (/ 1 2) + x1 + x2
n__terms(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: x2
 #add(x1,x2)	weight: 0
   add(x1,x2)	weight: (/ 1 2)
   sqr(x1)	weight: (/ 1 4) + x1
 terms(x1)	weight: (/ 1 4) + x1
    Usable rules: { 1 8..16 }
    Removed DPs: #2 #3 #16..18
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 4) + x1
n__first(x1,x2)	weight: (/ 1 2) + x1 + x2
 recip(x1)	weight: (/ 1 4)
activate(x1)	weight: x1
   dbl(x1)	weight: (/ 1 4) + x1
 #dbl(x1)	weight: 0
#terms(x1)	weight: 0
#activate(x1)	weight: x1
  n__s(x1)	weight: (/ 1 4) + x1
 #sqr(x1)	weight: 0
     0()	weight: 0
   #s(x1)	weight: 0
#first(x1,x2)	weight: (/ 1 4) + x2
   nil()	weight: 0
 first(x1,x2)	weight: (/ 1 2) + x1 + x2
n__terms(x1)	weight: (/ 1 4) + x1
  cons(x1,x2)	weight: 0
 #add(x1,x2)	weight: 0
   add(x1,x2)	weight: (/ 1 2)
   sqr(x1)	weight: (/ 1 4) + x1
 terms(x1)	weight: (/ 1 4) + x1
    Usable rules: { }
    Removed DPs: #5
Number of SCCs: 0, DPs: 0, edges: 0
YES