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
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [activate#](x0) = x0,
      
      [first#](x0, x1) = x1 + 0,
      
      [half](x0) = 1x0,
      
      [n__first](x0, x1) = 1x0 + 1x1 + 7,
      
      [activate](x0) = x0 + 6,
      
      [nil] = 1,
      
      [first](x0, x1) = 1x0 + 1x1 + 7,
      
      [add](x0, x1) = x1 + 6,
      
      [dbl](x0) = x0 + 6,
      
      [s](x0) = x0 + 6,
      
      [0] = 1,
      
      [cons](x0, x1) = x0 + x1 + 2,
      
      [n__terms](x0) = x0 + 4,
      
      [n__s](x0) = x0 + 6,
      
      [recip](x0) = x0 + 4,
      
      [sqr](x0) = x0 + 4,
      
      [terms](x0) = x0 + 6
     orientation:
      activate#(n__first(X1,X2)) = 1X1 + 1X2 + 7 >= X2 = activate#(X2)
      
      activate#(n__terms(X)) = X + 4 >= X = activate#(X)
      
      activate#(n__s(X)) = X + 6 >= X = activate#(X)
      
      activate#(n__first(X1,X2)) = 1X1 + 1X2 + 7 >= X1 = activate#(X1)
      
      activate#(n__first(X1,X2)) = 1X1 + 1X2 + 7 >= X2 + 6 = first#(activate(X1),activate(X2))
      
      first#(s(X),cons(Y,Z)) = Y + Z + 2 >= Z = activate#(Z)
      
      terms(N) = N + 6 >= N + 6 = cons(recip(sqr(N)),n__terms(n__s(N)))
      
      sqr(0()) = 4 >= 1 = 0()
      
      sqr(s(X)) = X + 6 >= X + 6 = s(add(sqr(X),dbl(X)))
      
      dbl(0()) = 6 >= 1 = 0()
      
      dbl(s(X)) = X + 6 >= X + 6 = s(s(dbl(X)))
      
      add(0(),X) = X + 6 >= X = X
      
      add(s(X),Y) = Y + 6 >= Y + 6 = s(add(X,Y))
      
      first(0(),X) = 1X + 7 >= 1 = nil()
      
      first(s(X),cons(Y,Z)) = 1X + 1Y + 1Z + 7 >= 1X + Y + 1Z + 7 = cons(Y,n__first(X,activate(Z)))
      
      half(0()) = 2 >= 1 = 0()
      
      half(s(0())) = 7 >= 1 = 0()
      
      half(s(s(X))) = 1X + 7 >= 1X + 6 = s(half(X))
      
      half(dbl(X)) = 1X + 7 >= X = X
      
      terms(X) = X + 6 >= X + 4 = n__terms(X)
      
      s(X) = X + 6 >= X + 6 = n__s(X)
      
      first(X1,X2) = 1X1 + 1X2 + 7 >= 1X1 + 1X2 + 7 = n__first(X1,X2)
      
      activate(n__terms(X)) = X + 6 >= X + 6 = terms(activate(X))
      
      activate(n__s(X)) = X + 6 >= X + 6 = s(activate(X))
      
      activate(n__first(X1,X2)) = 1X1 + 1X2 + 7 >= 1X1 + 1X2 + 7 = first(activate(X1),activate(X2))
      
      activate(X) = X + 6 >= X = X
     problem:
      DPs:
       activate#(n__terms(X)) -> activate#(X)
       activate#(n__s(X)) -> activate#(X)
       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
     SCC Processor:
      #sccs: 1
      #rules: 2
      #arcs: 26/9
      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