MAYBE

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

Proof:
 DP Processor:
  DPs:
   terms#(N) -> s#(N)
   terms#(N) -> sqr#(N)
   sqr#(s(X)) -> dbl#(activate(X))
   sqr#(s(X)) -> activate#(X)
   sqr#(s(X)) -> sqr#(activate(X))
   sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
   dbl#(s(X)) -> activate#(X)
   dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
   add#(s(X),Y) -> activate#(X)
   add#(s(X),Y) -> s#(n__add(activate(X),Y))
   first#(s(X),cons(Y,Z)) -> activate#(Z)
   first#(s(X),cons(Y,Z)) -> activate#(X)
   activate#(n__terms(X)) -> terms#(X)
   activate#(n__add(X1,X2)) -> add#(X1,X2)
   activate#(n__s(X)) -> s#(X)
   activate#(n__dbl(X)) -> dbl#(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(n__add(sqr(activate(X)),dbl(activate(X))))
   dbl(0()) -> 0()
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0(),X) -> X
   add(s(X),Y) -> s(n__add(activate(X),Y))
   first(0(),X) -> nil()
   first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
   terms(X) -> n__terms(X)
   add(X1,X2) -> n__add(X1,X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1,X2) -> n__first(X1,X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1,X2)) -> add(X1,X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1,X2)) -> first(X1,X2)
   activate(X) -> X
  TDG Processor:
   DPs:
    terms#(N) -> s#(N)
    terms#(N) -> sqr#(N)
    sqr#(s(X)) -> dbl#(activate(X))
    sqr#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X))
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    dbl#(s(X)) -> activate#(X)
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    add#(s(X),Y) -> activate#(X)
    add#(s(X),Y) -> s#(n__add(activate(X),Y))
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    first#(s(X),cons(Y,Z)) -> activate#(X)
    activate#(n__terms(X)) -> terms#(X)
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    activate#(n__s(X)) -> s#(X)
    activate#(n__dbl(X)) -> dbl#(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(n__add(sqr(activate(X)),dbl(activate(X))))
    dbl(0()) -> 0()
    dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
    add(0(),X) -> X
    add(s(X),Y) -> s(n__add(activate(X),Y))
    first(0(),X) -> nil()
    first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
    terms(X) -> n__terms(X)
    add(X1,X2) -> n__add(X1,X2)
    s(X) -> n__s(X)
    dbl(X) -> n__dbl(X)
    first(X1,X2) -> n__first(X1,X2)
    activate(n__terms(X)) -> terms(X)
    activate(n__add(X1,X2)) -> add(X1,X2)
    activate(n__s(X)) -> s(X)
    activate(n__dbl(X)) -> dbl(X)
    activate(n__first(X1,X2)) -> first(X1,X2)
    activate(X) -> X
   graph:
    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__dbl(X)) -> dbl#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__terms(X)) -> terms#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__dbl(X)) -> dbl#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(X)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    first#(s(X),cons(Y,Z)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    add#(s(X),Y) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    add#(s(X),Y) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__add(X1,X2)) -> add#(X1,X2)
    add#(s(X),Y) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(X)
    activate#(n__first(X1,X2)) -> first#(X1,X2) ->
    first#(s(X),cons(Y,Z)) -> activate#(Z)
    activate#(n__dbl(X)) -> dbl#(X) ->
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    activate#(n__dbl(X)) -> dbl#(X) ->
    dbl#(s(X)) -> activate#(X)
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> s#(n__add(activate(X),Y))
    activate#(n__add(X1,X2)) -> add#(X1,X2) ->
    add#(s(X),Y) -> activate#(X)
    activate#(n__terms(X)) -> terms#(X) -> terms#(N) -> sqr#(N)
    activate#(n__terms(X)) -> terms#(X) -> terms#(N) -> s#(N)
    dbl#(s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    dbl#(s(X)) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    dbl#(s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    dbl#(s(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(X1,X2)
    dbl#(s(X)) -> activate#(X) -> activate#(n__terms(X)) -> terms#(X)
    sqr#(s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> first#(X1,X2)
    sqr#(s(X)) -> activate#(X) -> activate#(n__dbl(X)) -> dbl#(X)
    sqr#(s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(X)
    sqr#(s(X)) -> activate#(X) -> activate#(n__add(X1,X2)) -> add#(X1,X2)
    sqr#(s(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(X)
    sqr#(s(X)) -> dbl#(activate(X)) ->
    dbl#(s(X)) -> s#(n__s(n__dbl(activate(X))))
    sqr#(s(X)) -> dbl#(activate(X)) -> dbl#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X)) ->
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    sqr#(s(X)) -> sqr#(activate(X)) ->
    sqr#(s(X)) -> sqr#(activate(X))
    sqr#(s(X)) -> sqr#(activate(X)) -> sqr#(s(X)) -> activate#(X)
    sqr#(s(X)) -> sqr#(activate(X)) -> sqr#(s(X)) -> dbl#(activate(X))
    terms#(N) -> sqr#(N) ->
    sqr#(s(X)) -> s#(n__add(sqr(activate(X)),dbl(activate(X))))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> sqr#(activate(X))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> activate#(X)
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> dbl#(activate(X))
   SCC Processor:
    #sccs: 1
    #rules: 12
    #arcs: 43/289
    DPs:
     first#(s(X),cons(Y,Z)) -> activate#(Z)
     activate#(n__terms(X)) -> terms#(X)
     terms#(N) -> sqr#(N)
     sqr#(s(X)) -> dbl#(activate(X))
     dbl#(s(X)) -> activate#(X)
     activate#(n__add(X1,X2)) -> add#(X1,X2)
     add#(s(X),Y) -> activate#(X)
     activate#(n__dbl(X)) -> dbl#(X)
     activate#(n__first(X1,X2)) -> first#(X1,X2)
     first#(s(X),cons(Y,Z)) -> activate#(X)
     sqr#(s(X)) -> activate#(X)
     sqr#(s(X)) -> sqr#(activate(X))
    TRS:
     terms(N) -> cons(recip(sqr(N)),n__terms(s(N)))
     sqr(0()) -> 0()
     sqr(s(X)) -> s(n__add(sqr(activate(X)),dbl(activate(X))))
     dbl(0()) -> 0()
     dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
     add(0(),X) -> X
     add(s(X),Y) -> s(n__add(activate(X),Y))
     first(0(),X) -> nil()
     first(s(X),cons(Y,Z)) -> cons(Y,n__first(activate(X),activate(Z)))
     terms(X) -> n__terms(X)
     add(X1,X2) -> n__add(X1,X2)
     s(X) -> n__s(X)
     dbl(X) -> n__dbl(X)
     first(X1,X2) -> n__first(X1,X2)
     activate(n__terms(X)) -> terms(X)
     activate(n__add(X1,X2)) -> add(X1,X2)
     activate(n__s(X)) -> s(X)
     activate(n__dbl(X)) -> dbl(X)
     activate(n__first(X1,X2)) -> first(X1,X2)
     activate(X) -> X
    Open