(STRATEGY
    INNERMOST)

(VAR
    N X XS YS ZS)
(DATATYPES
    A = µX.< cons(X, X), n__natsFrom(X), n__s(X), pair(X, X), 0, nil >)
(SIGNATURES
    natsFrom :: [A] -> A
    fst :: [A] -> A
    snd :: [A] -> A
    splitAt :: [A x A] -> A
    u :: [A x A x A x A] -> A
    head :: [A] -> A
    tail :: [A] -> A
    sel :: [A x A] -> A
    take :: [A x A] -> A
    afterNth :: [A x A] -> A
    s :: [A] -> A
    activate :: [A] -> A)
(RULES
    natsFrom(N) -> cons(N
                       ,n__natsFrom(n__s(N)))
    fst(pair(XS,YS)) -> XS
    snd(pair(XS,YS)) -> YS
    splitAt(0(),XS) -> pair(nil()
                           ,XS)
    splitAt(s(N),cons(X,XS)) ->
      u(splitAt(N,activate(XS))
       ,N
       ,X
       ,activate(XS))
    u(pair(YS,ZS),N,X,XS) ->
      pair(cons(activate(X),YS),ZS)
    head(cons(N,XS)) -> N
    tail(cons(N,XS)) -> activate(XS)
    sel(N,XS) -> head(afterNth(N
                              ,XS))
    take(N,XS) -> fst(splitAt(N,XS))
    afterNth(N,XS) -> snd(splitAt(N
                                 ,XS))
    natsFrom(X) -> n__natsFrom(X)
    s(X) -> n__s(X)
    activate(n__natsFrom(X)) ->
      natsFrom(activate(X))
    activate(n__s(X)) ->
      s(activate(X))
    activate(X) -> X)