YES

Problem:
 terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
 s(X) -> n__s(X)
 first(X1,X2) -> n__first(X1,X2)
 activate(n__terms(X)) -> terms(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(n__first(X1,X2)) -> first(activate(X1),activate(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))
   sqr#(s(X)) -> s#(add(sqr(X),dbl(X)))
   dbl#(s(X)) -> dbl#(X)
   dbl#(s(X)) -> s#(dbl(X))
   dbl#(s(X)) -> s#(s(dbl(X)))
   add#(s(X),Y) -> add#(X,Y)
   add#(s(X),Y) -> s#(add(X,Y))
   first#(s(X),cons(Y,Z)) -> activate#(Z)
   half#(s(s(X))) -> half#(X)
   half#(s(s(X))) -> s#(half(X))
   activate#(n__terms(X)) -> activate#(X)
   activate#(n__terms(X)) -> terms#(activate(X))
   activate#(n__s(X)) -> activate#(X)
   activate#(n__s(X)) -> s#(activate(X))
   activate#(n__first(X1,X2)) -> activate#(X2)
   activate#(n__first(X1,X2)) -> activate#(X1)
   activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
  TRS:
   terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
   s(X) -> n__s(X)
   first(X1,X2) -> n__first(X1,X2)
   activate(n__terms(X)) -> terms(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1,X2)) -> first(activate(X1),activate(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))
    sqr#(s(X)) -> s#(add(sqr(X),dbl(X)))
    dbl#(s(X)) -> dbl#(X)
    dbl#(s(X)) -> s#(dbl(X))
    dbl#(s(X)) -> s#(s(dbl(X)))
    add#(s(X),Y) -> add#(X,Y)
    add#(s(X),Y) -> s#(add(X,Y))
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    half#(s(s(X))) -> half#(X)
    half#(s(s(X))) -> s#(half(X))
    activate#(n__terms(X)) -> activate#(X)
    activate#(n__terms(X)) -> terms#(activate(X))
    activate#(n__s(X)) -> activate#(X)
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
   TRS:
    terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
    s(X) -> n__s(X)
    first(X1,X2) -> n__first(X1,X2)
    activate(n__terms(X)) -> terms(activate(X))
    activate(n__s(X)) -> s(activate(X))
    activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
    activate(X) -> X
   graph:
    half#(s(s(X))) -> half#(X) -> half#(s(s(X))) -> s#(half(X))
    half#(s(s(X))) -> half#(X) ->
    half#(s(s(X))) -> half#(X)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__terms(X)) -> terms#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__terms(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__terms(X)) -> terms#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__terms(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2)) ->
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(activate(X))
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> activate#(X)
    activate#(n__terms(X)) -> terms#(activate(X)) ->
    terms#(N) -> sqr#(N)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(activate(X))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> activate#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> activate#(X1)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__first(X1,X2)) -> activate#(X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> s#(activate(X))
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> activate#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__terms(X)) -> terms#(activate(X))
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__terms(X)) -> activate#(X)
    add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> s#(add(X,Y))
    add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> add#(X,Y)
    dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(s(dbl(X)))
    dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(dbl(X))
    dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X)
    sqr#(s(X)) -> add#(sqr(X),dbl(X)) ->
    add#(s(X),Y) -> s#(add(X,Y))
    sqr#(s(X)) -> add#(sqr(X),dbl(X)) -> add#(s(X),Y) -> add#(X,Y)
    sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(s(dbl(X)))
    sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(dbl(X))
    sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X)
    sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> s#(add(sqr(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)) -> s#(add(sqr(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: 10
    #arcs: 57/400
    DPs:
     activate#(n__first(X1,X2)) -> activate#(X2)
     activate#(n__terms(X)) -> activate#(X)
     activate#(n__s(X)) -> activate#(X)
     activate#(n__first(X1,X2)) -> activate#(X1)
     activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
     first#(s(X),cons(Y,Z)) -> activate#(Z)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
     s(X) -> n__s(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(activate(X))
     activate(n__s(X)) -> s(activate(X))
     activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
     activate(X) -> X
    Usable Rule Processor:
     DPs:
      activate#(n__first(X1,X2)) -> activate#(X2)
      activate#(n__terms(X)) -> activate#(X)
      activate#(n__s(X)) -> activate#(X)
      activate#(n__first(X1,X2)) -> activate#(X1)
      activate#(n__first(X1,X2)) -> first#(activate(X1),activate(X2))
      first#(s(X),cons(Y,Z)) -> activate#(Z)
     TRS:
      activate(n__terms(X)) -> terms(activate(X))
      activate(n__s(X)) -> s(activate(X))
      activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
      activate(X) -> X
      terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
      terms(X) -> n__terms(X)
      s(X) -> n__s(X)
      first(0(),X) -> nil()
      first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
      first(X1,X2) -> n__first(X1,X2)
      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))
     KBO Processor:
      argument filtering:
       pi(terms) = 0
       pi(sqr) = []
       pi(recip) = 0
       pi(n__s) = 0
       pi(n__terms) = 0
       pi(cons) = 1
       pi(0) = []
       pi(s) = 0
       pi(dbl) = []
       pi(add) = [0,1]
       pi(first) = [0,1]
       pi(nil) = []
       pi(activate) = [0]
       pi(n__first) = [0,1]
       pi(first#) = [1]
       pi(activate#) = 0
      usable rules:
       activate(n__terms(X)) -> terms(activate(X))
       activate(n__s(X)) -> s(activate(X))
       activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
       activate(X) -> X
       terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
       terms(X) -> n__terms(X)
       s(X) -> n__s(X)
       first(0(),X) -> nil()
       first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
       first(X1,X2) -> n__first(X1,X2)
      weight function:
       w0 = 1
       w(n__first) = w(nil) = w(first) = w(add) = w(dbl) = w(s) = w(0) = w(
       recip) = w(sqr) = 1
       w(activate#) = w(first#) = w(activate) = w(cons) = w(n__terms) = w(
       n__s) = w(terms) = 0
      precedence:
       recip > first# ~ activate ~ nil ~ add ~ dbl ~ n__s > first > activate# ~ 
       n__first ~ s ~ 0 ~ cons ~ n__terms ~ sqr ~ terms
      problem:
       DPs:
        activate#(n__terms(X)) -> activate#(X)
        activate#(n__s(X)) -> activate#(X)
       TRS:
        activate(n__terms(X)) -> terms(activate(X))
        activate(n__s(X)) -> s(activate(X))
        activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
        activate(X) -> X
        terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
        terms(X) -> n__terms(X)
        s(X) -> n__s(X)
        first(0(),X) -> nil()
        first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
        first(X1,X2) -> n__first(X1,X2)
        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))
      Restore Modifier:
       DPs:
        activate#(n__terms(X)) -> activate#(X)
        activate#(n__s(X)) -> activate#(X)
       TRS:
        terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
        s(X) -> n__s(X)
        first(X1,X2) -> n__first(X1,X2)
        activate(n__terms(X)) -> terms(activate(X))
        activate(n__s(X)) -> s(activate(X))
        activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
        activate(X) -> X
       Subterm Criterion Processor:
        simple projection:
         pi(activate#) = 0
        problem:
         DPs:
          
         TRS:
          terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
          s(X) -> n__s(X)
          first(X1,X2) -> n__first(X1,X2)
          activate(n__terms(X)) -> terms(activate(X))
          activate(n__s(X)) -> s(activate(X))
          activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
          activate(X) -> X
        Qed
    
    DPs:
     sqr#(s(X)) -> sqr#(X)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
     s(X) -> n__s(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(activate(X))
     activate(n__s(X)) -> s(activate(X))
     activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
     activate(X) -> X
    Subterm Criterion Processor:
     simple projection:
      pi(sqr#) = 0
     problem:
      DPs:
       
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
       s(X) -> n__s(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(activate(X))
       activate(n__s(X)) -> s(activate(X))
       activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
       activate(X) -> X
     Qed
    
    DPs:
     add#(s(X),Y) -> add#(X,Y)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
     s(X) -> n__s(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(activate(X))
     activate(n__s(X)) -> s(activate(X))
     activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
     activate(X) -> X
    Subterm Criterion Processor:
     simple projection:
      pi(add#) = 0
     problem:
      DPs:
       
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
       s(X) -> n__s(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(activate(X))
       activate(n__s(X)) -> s(activate(X))
       activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
       activate(X) -> X
     Qed
    
    DPs:
     dbl#(s(X)) -> dbl#(X)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
     s(X) -> n__s(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(activate(X))
     activate(n__s(X)) -> s(activate(X))
     activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
     activate(X) -> X
    Subterm Criterion Processor:
     simple projection:
      pi(dbl#) = 0
     problem:
      DPs:
       
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
       s(X) -> n__s(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(activate(X))
       activate(n__s(X)) -> s(activate(X))
       activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
       activate(X) -> X
     Qed
    
    DPs:
     half#(s(s(X))) -> half#(X)
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
     s(X) -> n__s(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(activate(X))
     activate(n__s(X)) -> s(activate(X))
     activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
     activate(X) -> X
    Subterm Criterion Processor:
     simple projection:
      pi(half#) = 0
     problem:
      DPs:
       
      TRS:
       terms(N) -> cons(recip(sqr(N)),n__terms(n__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)
       s(X) -> n__s(X)
       first(X1,X2) -> n__first(X1,X2)
       activate(n__terms(X)) -> terms(activate(X))
       activate(n__s(X)) -> s(activate(X))
       activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
       activate(X) -> X
     Qed