(STRATEGY
    INNERMOST)

(VAR
    N V1 V2 X X1 X2 X3 X4 XS Y YS
    ZS)
(DATATYPES
    A = µX.< tt, cons(X, X), natsFrom(X), s(X), pair(X, X), nil, isLNat(X), afterNth(X, X), fst(X), snd(X), tail(X), take(X, X), 0, head(X), sel(X, X), splitAt(X, X), and(X, X), isNatural(X), U101(X, X, X), U11(X, X, X), U21(X, X), U31(X, X), U41(X, X), U51(X, X, X), U61(X, X), U71(X, X), U81(X, X, X, X), U82(X, X), U91(X, X), isPLNat(X) >)
(SIGNATURES
    a__U101 :: [A x A x A] -> A
    a__U11 :: [A x A x A] -> A
    a__U21 :: [A x A] -> A
    a__U31 :: [A x A] -> A
    a__U41 :: [A x A] -> A
    a__U51 :: [A x A x A] -> A
    a__U61 :: [A x A] -> A
    a__U71 :: [A x A] -> A
    a__U81 :: [A x A x A x A] -> A
    a__U82 :: [A x A] -> A
    a__U91 :: [A x A] -> A
    a__afterNth :: [A x A] -> A
    a__and :: [A x A] -> A
    a__fst :: [A] -> A
    a__head :: [A] -> A
    a__isLNat :: [A] -> A
    a__isNatural :: [A] -> A
    a__isPLNat :: [A] -> A
    a__natsFrom :: [A] -> A
    a__sel :: [A x A] -> A
    a__snd :: [A] -> A
    a__splitAt :: [A x A] -> A
    a__tail :: [A] -> A
    a__take :: [A x A] -> A
    mark :: [A] -> A)
(RULES
    a__U101(tt(),N,XS) ->
      a__fst(a__splitAt(mark(N)
                       ,mark(XS)))
    a__U11(tt(),N,XS) ->
      a__snd(a__splitAt(mark(N)
                       ,mark(XS)))
    a__U21(tt(),X) -> mark(X)
    a__U31(tt(),N) -> mark(N)
    a__U41(tt(),N) -> cons(mark(N)
                          ,natsFrom(s(N)))
    a__U51(tt(),N,XS) ->
      a__head(a__afterNth(mark(N)
                         ,mark(XS)))
    a__U61(tt(),Y) -> mark(Y)
    a__U71(tt(),XS) -> pair(nil()
                           ,mark(XS))
    a__U81(tt(),N,X,XS) ->
      a__U82(a__splitAt(mark(N)
                       ,mark(XS))
            ,X)
    a__U82(pair(YS,ZS),X) ->
      pair(cons(mark(X),YS),mark(ZS))
    a__U91(tt(),XS) -> mark(XS)
    a__afterNth(N,XS) ->
      a__U11(a__and(a__isNatural(N)
                   ,isLNat(XS))
            ,N
            ,XS)
    a__and(tt(),X) -> mark(X)
    a__fst(pair(X,Y)) ->
      a__U21(a__and(a__isLNat(X)
                   ,isLNat(Y))
            ,X)
    a__head(cons(N,XS)) ->
      a__U31(a__and(a__isNatural(N)
                   ,isLNat(XS))
            ,N)
    a__isLNat(nil()) -> tt()
    a__isLNat(afterNth(V1,V2)) ->
      a__and(a__isNatural(V1)
            ,isLNat(V2))
    a__isLNat(cons(V1,V2)) ->
      a__and(a__isNatural(V1)
            ,isLNat(V2))
    a__isLNat(fst(V1)) ->
      a__isPLNat(V1)
    a__isLNat(natsFrom(V1)) ->
      a__isNatural(V1)
    a__isLNat(snd(V1)) ->
      a__isPLNat(V1)
    a__isLNat(tail(V1)) ->
      a__isLNat(V1)
    a__isLNat(take(V1,V2)) ->
      a__and(a__isNatural(V1)
            ,isLNat(V2))
    a__isNatural(0()) -> tt()
    a__isNatural(head(V1)) ->
      a__isLNat(V1)
    a__isNatural(s(V1)) ->
      a__isNatural(V1)
    a__isNatural(sel(V1,V2)) ->
      a__and(a__isNatural(V1)
            ,isLNat(V2))
    a__isPLNat(pair(V1,V2)) ->
      a__and(a__isLNat(V1),isLNat(V2))
    a__isPLNat(splitAt(V1,V2)) ->
      a__and(a__isNatural(V1)
            ,isLNat(V2))
    a__natsFrom(N) ->
      a__U41(a__isNatural(N),N)
    a__sel(N,XS) ->
      a__U51(a__and(a__isNatural(N)
                   ,isLNat(XS))
            ,N
            ,XS)
    a__snd(pair(X,Y)) ->
      a__U61(a__and(a__isLNat(X)
                   ,isLNat(Y))
            ,Y)
    a__splitAt(0(),XS) ->
      a__U71(a__isLNat(XS),XS)
    a__splitAt(s(N),cons(X,XS)) ->
      a__U81(a__and(a__isNatural(N)
                   ,and(isNatural(X),isLNat(XS)))
            ,N
            ,X
            ,XS)
    a__tail(cons(N,XS)) ->
      a__U91(a__and(a__isNatural(N)
                   ,isLNat(XS))
            ,XS)
    a__take(N,XS) ->
      a__U101(a__and(a__isNatural(N)
                    ,isLNat(XS))
             ,N
             ,XS)
    mark(U101(X1,X2,X3)) ->
      a__U101(mark(X1),X2,X3)
    mark(fst(X)) -> a__fst(mark(X))
    mark(splitAt(X1,X2)) ->
      a__splitAt(mark(X1),mark(X2))
    mark(U11(X1,X2,X3)) ->
      a__U11(mark(X1),X2,X3)
    mark(snd(X)) -> a__snd(mark(X))
    mark(U21(X1,X2)) ->
      a__U21(mark(X1),X2)
    mark(U31(X1,X2)) ->
      a__U31(mark(X1),X2)
    mark(U41(X1,X2)) ->
      a__U41(mark(X1),X2)
    mark(natsFrom(X)) ->
      a__natsFrom(mark(X))
    mark(U51(X1,X2,X3)) ->
      a__U51(mark(X1),X2,X3)
    mark(head(X)) ->
      a__head(mark(X))
    mark(afterNth(X1,X2)) ->
      a__afterNth(mark(X1),mark(X2))
    mark(U61(X1,X2)) ->
      a__U61(mark(X1),X2)
    mark(U71(X1,X2)) ->
      a__U71(mark(X1),X2)
    mark(U81(X1,X2,X3,X4)) ->
      a__U81(mark(X1),X2,X3,X4)
    mark(U82(X1,X2)) ->
      a__U82(mark(X1),X2)
    mark(U91(X1,X2)) ->
      a__U91(mark(X1),X2)
    mark(and(X1,X2)) ->
      a__and(mark(X1),X2)
    mark(isNatural(X)) ->
      a__isNatural(X)
    mark(isLNat(X)) -> a__isLNat(X)
    mark(isPLNat(X)) ->
      a__isPLNat(X)
    mark(tail(X)) ->
      a__tail(mark(X))
    mark(take(X1,X2)) ->
      a__take(mark(X1),mark(X2))
    mark(sel(X1,X2)) ->
      a__sel(mark(X1),mark(X2))
    mark(tt()) -> tt()
    mark(cons(X1,X2)) ->
      cons(mark(X1),X2)
    mark(s(X)) -> s(mark(X))
    mark(pair(X1,X2)) ->
      pair(mark(X1),mark(X2))
    mark(nil()) -> nil()
    mark(0()) -> 0()
    a__U101(X1,X2,X3) -> U101(X1
                             ,X2
                             ,X3)
    a__fst(X) -> fst(X)
    a__splitAt(X1,X2) -> splitAt(X1
                                ,X2)
    a__U11(X1,X2,X3) -> U11(X1
                           ,X2
                           ,X3)
    a__snd(X) -> snd(X)
    a__U21(X1,X2) -> U21(X1,X2)
    a__U31(X1,X2) -> U31(X1,X2)
    a__U41(X1,X2) -> U41(X1,X2)
    a__natsFrom(X) -> natsFrom(X)
    a__U51(X1,X2,X3) -> U51(X1
                           ,X2
                           ,X3)
    a__head(X) -> head(X)
    a__afterNth(X1,X2) ->
      afterNth(X1,X2)
    a__U61(X1,X2) -> U61(X1,X2)
    a__U71(X1,X2) -> U71(X1,X2)
    a__U81(X1,X2,X3,X4) -> U81(X1
                              ,X2
                              ,X3
                              ,X4)
    a__U82(X1,X2) -> U82(X1,X2)
    a__U91(X1,X2) -> U91(X1,X2)
    a__and(X1,X2) -> and(X1,X2)
    a__isNatural(X) -> isNatural(X)
    a__isLNat(X) -> isLNat(X)
    a__isPLNat(X) -> isPLNat(X)
    a__tail(X) -> tail(X)
    a__take(X1,X2) -> take(X1,X2)
    a__sel(X1,X2) -> sel(X1,X2))