YES

Problem:
 terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
 sqr(0()) -> 0()
 sqr(s(X)) -> s(add(sqr(X),dbl(X)))
 dbl(0()) -> 0()
 dbl(s(X)) -> s(s(dbl(X)))
 add(0(),X) -> X
 add(s(X),Y) -> s(add(X,Y))
 first(0(),X) -> nil()
 first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(X))) -> s(half(X))
 half(dbl(X)) -> X
 terms(X) -> n__terms(X)
 first(X1,X2) -> n__first(X1,X2)
 activate(n__terms(X)) -> terms(X)
 activate(n__first(X1,X2)) -> first(X1,X2)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   terms#(N) -> sqr#(N)
   sqr#(s(X)) -> dbl#(X)
   sqr#(s(X)) -> sqr#(X)
   sqr#(s(X)) -> add#(sqr(X),dbl(X))
   dbl#(s(X)) -> dbl#(X)
   add#(s(X),Y) -> add#(X,Y)
   first#(s(X),cons(Y,Z)) -> activate#(Z)
   half#(s(s(X))) -> half#(X)
   activate#(n__terms(X)) -> terms#(X)
   activate#(n__first(X1,X2)) -> first#(X1,X2)
  TRS:
   terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
   sqr(0()) -> 0()
   sqr(s(X)) -> s(add(sqr(X),dbl(X)))
   dbl(0()) -> 0()
   dbl(s(X)) -> s(s(dbl(X)))
   add(0(),X) -> X
   add(s(X),Y) -> s(add(X,Y))
   first(0(),X) -> nil()
   first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(X))) -> s(half(X))
   half(dbl(X)) -> X
   terms(X) -> n__terms(X)
   first(X1,X2) -> n__first(X1,X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__first(X1,X2)) -> first(X1,X2)
   activate(X) -> X
  TDG Processor:
   DPs:
    terms#(N) -> sqr#(N)
    sqr#(s(X)) -> dbl#(X)
    sqr#(s(X)) -> sqr#(X)
    sqr#(s(X)) -> add#(sqr(X),dbl(X))
    dbl#(s(X)) -> dbl#(X)
    add#(s(X),Y) -> add#(X,Y)
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    half#(s(s(X))) -> half#(X)
    activate#(n__terms(X)) -> terms#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2)
   TRS:
    terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
    sqr(0()) -> 0()
    sqr(s(X)) -> s(add(sqr(X),dbl(X)))
    dbl(0()) -> 0()
    dbl(s(X)) -> s(s(dbl(X)))
    add(0(),X) -> X
    add(s(X),Y) -> s(add(X,Y))
    first(0(),X) -> nil()
    first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(X))) -> s(half(X))
    half(dbl(X)) -> X
    terms(X) -> n__terms(X)
    first(X1,X2) -> n__first(X1,X2)
    activate(n__terms(X)) -> terms(X)
    activate(n__first(X1,X2)) -> first(X1,X2)
    activate(X) -> X
   graph:
    half#(s(s(X))) -> half#(X) ->
    half#(s(s(X))) -> half#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    activate#(n__terms(X)) -> terms#(X) ->
    terms#(N) -> sqr#(N)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__terms(X)) -> terms#(X)
    add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> add#(X,Y)
    dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X)
    sqr#(s(X)) -> add#(sqr(X),dbl(X)) -> add#(s(X),Y) -> add#(X,Y)
    sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X)
    sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> add#(sqr(X),dbl(X))
    sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> sqr#(X)
    sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> dbl#(X)
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> add#(sqr(X),dbl(X))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> sqr#(X)
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> dbl#(X)
   SCC Processor:
    #sccs: 5
    #rules: 6
    #arcs: 15/100
    DPs:
     activate#(n__first(X1,X2)) -> first#(X1,X2)
     first#(s(X),cons(Y,Z)) -> activate#(Z)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
     sqr(0()) -> 0()
     sqr(s(X)) -> s(add(sqr(X),dbl(X)))
     dbl(0()) -> 0()
     dbl(s(X)) -> s(s(dbl(X)))
     add(0(),X) -> X
     add(s(X),Y) -> s(add(X,Y))
     first(0(),X) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(X))) -> s(half(X))
     half(dbl(X)) -> X
     terms(X) -> n__terms(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(X)
     activate(n__first(X1,X2)) -> first(X1,X2)
     activate(X) -> X
    KBO Processor:
     argument filtering:
      pi(terms) = 0
      pi(sqr) = [0]
      pi(recip) = []
      pi(s) = 0
      pi(n__terms) = 0
      pi(cons) = 1
      pi(0) = []
      pi(dbl) = [0]
      pi(add) = [1]
      pi(first) = [1]
      pi(nil) = []
      pi(activate) = [0]
      pi(n__first) = [1]
      pi(half) = 0
      pi(first#) = [1]
      pi(activate#) = 0
     weight function:
      w0 = 1
      w(activate#) = w(first#) = w(n__first) = w(nil) = w(first) = w(
      0) = w(cons) = w(recip) = w(sqr) = w(terms) = 1
      w(half) = w(activate) = w(add) = w(dbl) = w(n__terms) = w(s) = 0
     precedence:
      activate ~ add ~ dbl > first > n__first > activate# ~ first# ~ 
      half ~ nil ~ 0 ~ cons ~ n__terms ~ s ~ recip ~ sqr ~ terms
     problem:
      DPs:
       
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(add(sqr(X),dbl(X)))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(s(dbl(X)))
       add(0(),X) -> X
       add(s(X),Y) -> s(add(X,Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(X))) -> s(half(X))
       half(dbl(X)) -> X
       terms(X) -> n__terms(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
     Qed
    
    DPs:
     sqr#(s(X)) -> sqr#(X)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
     sqr(0()) -> 0()
     sqr(s(X)) -> s(add(sqr(X),dbl(X)))
     dbl(0()) -> 0()
     dbl(s(X)) -> s(s(dbl(X)))
     add(0(),X) -> X
     add(s(X),Y) -> s(add(X,Y))
     first(0(),X) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(X))) -> s(half(X))
     half(dbl(X)) -> X
     terms(X) -> n__terms(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(X)
     activate(n__first(X1,X2)) -> first(X1,X2)
     activate(X) -> X
    LPO Processor:
     argument filtering:
      pi(terms) = [0]
      pi(sqr) = [0]
      pi(recip) = []
      pi(s) = [0]
      pi(n__terms) = 0
      pi(cons) = 1
      pi(0) = []
      pi(dbl) = [0]
      pi(add) = [0,1]
      pi(first) = [0]
      pi(nil) = []
      pi(activate) = [0]
      pi(n__first) = 0
      pi(half) = 0
      pi(sqr#) = 0
     precedence:
      activate > sqr > first ~ add ~ dbl > terms > sqr# ~ half ~ n__first ~ 
      nil ~ 0 ~ cons ~ n__terms ~ s ~ recip
     problem:
      DPs:
       
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(add(sqr(X),dbl(X)))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(s(dbl(X)))
       add(0(),X) -> X
       add(s(X),Y) -> s(add(X,Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(X))) -> s(half(X))
       half(dbl(X)) -> X
       terms(X) -> n__terms(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
     Qed
    
    DPs:
     add#(s(X),Y) -> add#(X,Y)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
     sqr(0()) -> 0()
     sqr(s(X)) -> s(add(sqr(X),dbl(X)))
     dbl(0()) -> 0()
     dbl(s(X)) -> s(s(dbl(X)))
     add(0(),X) -> X
     add(s(X),Y) -> s(add(X,Y))
     first(0(),X) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(X))) -> s(half(X))
     half(dbl(X)) -> X
     terms(X) -> n__terms(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(X)
     activate(n__first(X1,X2)) -> first(X1,X2)
     activate(X) -> X
    LPO Processor:
     argument filtering:
      pi(terms) = [0]
      pi(sqr) = [0]
      pi(recip) = []
      pi(s) = [0]
      pi(n__terms) = 0
      pi(cons) = 1
      pi(0) = []
      pi(dbl) = [0]
      pi(add) = [0,1]
      pi(first) = [0]
      pi(nil) = []
      pi(activate) = [0]
      pi(n__first) = 0
      pi(half) = 0
      pi(add#) = 0
     precedence:
      activate > first > sqr > add > dbl ~ terms > add# ~ half ~ n__first ~ 
      nil ~ 0 ~ cons ~ n__terms ~ s ~ recip
     problem:
      DPs:
       
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(add(sqr(X),dbl(X)))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(s(dbl(X)))
       add(0(),X) -> X
       add(s(X),Y) -> s(add(X,Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(X))) -> s(half(X))
       half(dbl(X)) -> X
       terms(X) -> n__terms(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
     Qed
    
    DPs:
     dbl#(s(X)) -> dbl#(X)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
     sqr(0()) -> 0()
     sqr(s(X)) -> s(add(sqr(X),dbl(X)))
     dbl(0()) -> 0()
     dbl(s(X)) -> s(s(dbl(X)))
     add(0(),X) -> X
     add(s(X),Y) -> s(add(X,Y))
     first(0(),X) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(X))) -> s(half(X))
     half(dbl(X)) -> X
     terms(X) -> n__terms(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(X)
     activate(n__first(X1,X2)) -> first(X1,X2)
     activate(X) -> X
    LPO Processor:
     argument filtering:
      pi(terms) = [0]
      pi(sqr) = [0]
      pi(recip) = []
      pi(s) = [0]
      pi(n__terms) = 0
      pi(cons) = 1
      pi(0) = []
      pi(dbl) = [0]
      pi(add) = [0,1]
      pi(first) = [0]
      pi(nil) = []
      pi(activate) = [0]
      pi(n__first) = 0
      pi(half) = 0
      pi(dbl#) = 0
     precedence:
      activate > sqr > first ~ add ~ dbl > terms > dbl# ~ half ~ n__first ~ 
      nil ~ 0 ~ cons ~ n__terms ~ s ~ recip
     problem:
      DPs:
       
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
       sqr(0()) -> 0()
       sqr(s(X)) -> s(add(sqr(X),dbl(X)))
       dbl(0()) -> 0()
       dbl(s(X)) -> s(s(dbl(X)))
       add(0(),X) -> X
       add(s(X),Y) -> s(add(X,Y))
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(X))) -> s(half(X))
       half(dbl(X)) -> X
       terms(X) -> n__terms(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(X)
       activate(n__first(X1,X2)) -> first(X1,X2)
       activate(X) -> X
     Qed
   
   DPs:
    half#(s(s(X))) -> half#(X)
   TRS:
    terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
    sqr(0()) -> 0()
    sqr(s(X)) -> s(add(sqr(X),dbl(X)))
    dbl(0()) -> 0()
    dbl(s(X)) -> s(s(dbl(X)))
    add(0(),X) -> X
    add(s(X),Y) -> s(add(X,Y))
    first(0(),X) -> nil()
    first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(X))) -> s(half(X))
    half(dbl(X)) -> X
    terms(X) -> n__terms(X)
    first(X1,X2) -> n__first(X1,X2)
    activate(n__terms(X)) -> terms(X)
    activate(n__first(X1,X2)) -> first(X1,X2)
    activate(X) -> X
   LPO Processor:
    argument filtering:
     pi(terms) = [0]
     pi(sqr) = [0]
     pi(recip) = []
     pi(s) = [0]
     pi(n__terms) = 0
     pi(cons) = 1
     pi(0) = []
     pi(dbl) = [0]
     pi(add) = [0,1]
     pi(first) = [0]
     pi(nil) = []
     pi(activate) = [0]
     pi(n__first) = 0
     pi(half) = 0
     pi(half#) = [0]
    precedence:
     activate > sqr > add ~ dbl ~ terms > s > first > half# ~ half ~ 
     n__first ~ nil ~ 0 ~ cons ~ n__terms ~ recip
    problem:
     DPs:
      
     TRS:
      terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
      sqr(0()) -> 0()
      sqr(s(X)) -> s(add(sqr(X),dbl(X)))
      dbl(0()) -> 0()
      dbl(s(X)) -> s(s(dbl(X)))
      add(0(),X) -> X
      add(s(X),Y) -> s(add(X,Y))
      first(0(),X) -> nil()
      first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
      half(0()) -> 0()
      half(s(0())) -> 0()
      half(s(s(X))) -> s(half(X))
      half(dbl(X)) -> X
      terms(X) -> n__terms(X)
      first(X1,X2) -> n__first(X1,X2)
      activate(n__terms(X)) -> terms(X)
      activate(n__first(X1,X2)) -> first(X1,X2)
      activate(X) -> X
    Qed