MAYBE

Problem:
 terms(N) -> cons(recip(sqr(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)) -> cons(Y)
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(X))) -> s(half(X))
 half(dbl(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)
   half#(s(s(X))) -> half#(X)
  TRS:
   terms(N) -> cons(recip(sqr(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)) -> cons(Y)
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(X))) -> s(half(X))
   half(dbl(X)) -> X
  Usable Rule 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)
    half#(s(s(X))) -> half#(X)
   TRS:
    f17(x,y) -> x
    f17(x,y) -> y
    dbl(0()) -> 0()
    dbl(s(X)) -> s(s(dbl(X)))
    sqr(0()) -> 0()
    sqr(s(X)) -> s(add(sqr(X),dbl(X)))
    add(0(),X) -> X
    add(s(X),Y) -> s(add(X,Y))
   CDG 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)
     half#(s(s(X))) -> half#(X)
    TRS:
     f17(x,y) -> x
     f17(x,y) -> y
     dbl(0()) -> 0()
     dbl(s(X)) -> s(s(dbl(X)))
     sqr(0()) -> 0()
     sqr(s(X)) -> s(add(sqr(X),dbl(X)))
     add(0(),X) -> X
     add(s(X),Y) -> s(add(X,Y))
    graph:
     half#(s(s(X))) -> half#(X) -> half#(s(s(X))) -> half#(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)) -> 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))
     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))
    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))
      dbl#(s(X)) -> dbl#(X)
      add#(s(X),Y) -> add#(X,Y)
      half#(s(s(X))) -> half#(X)
     TRS:
      terms(N) -> cons(recip(sqr(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)) -> cons(Y)
      half(0()) -> 0()
      half(s(0())) -> 0()
      half(s(s(X))) -> s(half(X))
      half(dbl(X)) -> X
     SCC Processor:
      #sccs: 4
      #rules: 4
      #arcs: 11/49
      DPs:
       sqr#(s(X)) -> sqr#(X)
      TRS:
       terms(N) -> cons(recip(sqr(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)) -> cons(Y)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(X))) -> s(half(X))
       half(dbl(X)) -> X
      Open
      
      DPs:
       dbl#(s(X)) -> dbl#(X)
      TRS:
       terms(N) -> cons(recip(sqr(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)) -> cons(Y)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(X))) -> s(half(X))
       half(dbl(X)) -> X
      Open
      
      DPs:
       add#(s(X),Y) -> add#(X,Y)
      TRS:
       terms(N) -> cons(recip(sqr(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)) -> cons(Y)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(X))) -> s(half(X))
       half(dbl(X)) -> X
      Open
      
      DPs:
       half#(s(s(X))) -> half#(X)
      TRS:
       terms(N) -> cons(recip(sqr(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)) -> cons(Y)
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(X))) -> s(half(X))
       half(dbl(X)) -> X
      Open