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