MAYBE

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)))
 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)
   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)))
   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
  EDG 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)
    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)))
    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:
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__terms(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__s(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X2) ->
    activate#(n__first(X1,X2)) -> 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)) -> first#(activate(X1),activate(X2))
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__terms(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__s(X)) -> activate#(X)
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__first(X1,X2)) -> activate#(X1) ->
    activate#(n__first(X1,X2)) -> 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)) -> first#(activate(X1),activate(X2))
    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__terms(X)) -> activate#(X)
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(activate(X))
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__terms(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> 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)) -> first#(activate(X1),activate(X2))
    activate#(n__terms(X)) -> terms#(activate(X)) ->
    terms#(N) -> sqr#(N)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> activate#(X)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__terms(X)) -> terms#(activate(X))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__s(X)) -> activate#(X)
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__s(X)) -> s#(activate(X))
    activate#(n__s(X)) -> activate#(X) ->
    activate#(n__first(X1,X2)) -> 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)) -> first#(activate(X1),activate(X2))
    first#(s(X),cons(Y,Z)) -> activate#(Z) ->
    activate#(n__terms(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__s(X)) -> activate#(X)
    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__first(X1,X2)) -> 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)) -> first#(activate(X1),activate(X2))
    add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> add#(X,Y)
    add#(s(X),Y) -> add#(X,Y) -> add#(s(X),Y) -> s#(add(X,Y))
    dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X)
    dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(dbl(X))
    dbl#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(s(dbl(X)))
    sqr#(s(X)) -> add#(sqr(X),dbl(X)) -> add#(s(X),Y) -> add#(X,Y)
    sqr#(s(X)) -> add#(sqr(X),dbl(X)) -> add#(s(X),Y) -> s#(add(X,Y))
    sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> dbl#(X)
    sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(dbl(X))
    sqr#(s(X)) -> dbl#(X) -> dbl#(s(X)) -> s#(s(dbl(X)))
    sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> dbl#(X)
    sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> sqr#(X)
    sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> add#(sqr(X),dbl(X))
    sqr#(s(X)) -> sqr#(X) -> sqr#(s(X)) -> s#(add(sqr(X),dbl(X)))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> dbl#(X)
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> sqr#(X)
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> add#(sqr(X),dbl(X))
    terms#(N) -> sqr#(N) -> sqr#(s(X)) -> s#(add(sqr(X),dbl(X)))
   Restore Modifier:
    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)
     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)))
     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: 4
     #rules: 9
     #arcs: 55/324
     DPs:
      activate#(n__first(X1,X2)) -> activate#(X2)
      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)
      activate#(n__s(X)) -> activate#(X)
      activate#(n__terms(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)))
      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
     Open
     
     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)))
      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
     Open
     
     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)))
      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
     Open
     
     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)))
      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
     Open