MAYBE MAYBE TRS: { U102(tt()) -> tt(), isLNat(n__natsFrom(V1)) -> U71(isNatural(activate(V1))), isLNat(n__nil()) -> tt(), isLNat(n__afterNth(V1, V2)) -> U41(isNatural(activate(V1)), activate(V2)), isLNat(n__cons(V1, V2)) -> U51(isNatural(activate(V1)), activate(V2)), isLNat(n__fst(V1)) -> U61(isPLNat(activate(V1))), isLNat(n__snd(V1)) -> U81(isPLNat(activate(V1))), isLNat(n__tail(V1)) -> U91(isLNat(activate(V1))), isLNat(n__take(V1, V2)) -> U101(isNatural(activate(V1)), activate(V2)), activate(X) -> X, activate(n__natsFrom(X)) -> natsFrom(activate(X)), activate(n__s(X)) -> s(activate(X)), activate(n__nil()) -> nil(), activate(n__afterNth(X1, X2)) -> afterNth(activate(X1), activate(X2)), activate(n__cons(X1, X2)) -> cons(activate(X1), X2), activate(n__fst(X)) -> fst(activate(X)), activate(n__snd(X)) -> snd(activate(X)), activate(n__tail(X)) -> tail(activate(X)), activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)), activate(n__0()) -> 0(), activate(n__head(X)) -> head(activate(X)), activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2)), activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2)), activate(n__splitAt(X1, X2)) -> splitAt(activate(X1), activate(X2)), U101(tt(), V2) -> U102(isLNat(activate(V2))), U12(tt(), N, XS) -> snd(splitAt(activate(N), activate(XS))), U11(tt(), N, XS) -> U12(isLNat(activate(XS)), activate(N), activate(XS)), U111(tt()) -> tt(), snd(X) -> n__snd(X), snd(pair(X, Y)) -> U181(isLNat(X), Y), splitAt(X1, X2) -> n__splitAt(X1, X2), splitAt(0(), XS) -> U191(isLNat(XS), XS), splitAt(s(N), cons(X, XS)) -> U201(isNatural(N), N, X, activate(XS)), U121(tt()) -> tt(), U132(tt()) -> tt(), U131(tt(), V2) -> U132(isLNat(activate(V2))), U142(tt()) -> tt(), U141(tt(), V2) -> U142(isLNat(activate(V2))), U152(tt()) -> tt(), U151(tt(), V2) -> U152(isLNat(activate(V2))), cons(X1, X2) -> n__cons(X1, X2), U161(tt(), N) -> cons(activate(N), n__natsFrom(n__s(activate(N)))), U172(tt(), N, XS) -> head(afterNth(activate(N), activate(XS))), U171(tt(), N, XS) -> U172(isLNat(activate(XS)), activate(N), activate(XS)), head(X) -> n__head(X), head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)), afterNth(N, XS) -> U11(isNatural(N), N, XS), afterNth(X1, X2) -> n__afterNth(X1, X2), U182(tt(), Y) -> activate(Y), U181(tt(), Y) -> U182(isLNat(activate(Y)), activate(Y)), pair(X1, X2) -> n__pair(X1, X2), nil() -> n__nil(), U191(tt(), XS) -> pair(nil(), activate(XS)), U202(tt(), N, X, XS) -> U203(isLNat(activate(XS)), activate(N), activate(X), activate(XS)), isNatural(n__s(V1)) -> U121(isNatural(activate(V1))), isNatural(n__0()) -> tt(), isNatural(n__head(V1)) -> U111(isLNat(activate(V1))), isNatural(n__sel(V1, V2)) -> U131(isNatural(activate(V1)), activate(V2)), U201(tt(), N, X, XS) -> U202(isNatural(activate(X)), activate(N), activate(X), activate(XS)), U203(tt(), N, X, XS) -> U204(splitAt(activate(N), activate(XS)), activate(X)), U204(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS), U22(tt(), X) -> activate(X), U21(tt(), X, Y) -> U22(isLNat(activate(Y)), activate(X)), U212(tt(), XS) -> activate(XS), U211(tt(), XS) -> U212(isLNat(activate(XS)), activate(XS)), U222(tt(), N, XS) -> fst(splitAt(activate(N), activate(XS))), U221(tt(), N, XS) -> U222(isLNat(activate(XS)), activate(N), activate(XS)), fst(X) -> n__fst(X), fst(pair(X, Y)) -> U21(isLNat(X), X, Y), U32(tt(), N) -> activate(N), U31(tt(), N, XS) -> U32(isLNat(activate(XS)), activate(N)), U42(tt()) -> tt(), U41(tt(), V2) -> U42(isLNat(activate(V2))), U52(tt()) -> tt(), U51(tt(), V2) -> U52(isLNat(activate(V2))), U61(tt()) -> tt(), U71(tt()) -> tt(), U81(tt()) -> tt(), U91(tt()) -> tt(), isPLNat(n__pair(V1, V2)) -> U141(isLNat(activate(V1)), activate(V2)), isPLNat(n__splitAt(V1, V2)) -> U151(isNatural(activate(V1)), activate(V2)), natsFrom(N) -> U161(isNatural(N), N), natsFrom(X) -> n__natsFrom(X), sel(N, XS) -> U171(isNatural(N), N, XS), sel(X1, X2) -> n__sel(X1, X2), 0() -> n__0(), s(X) -> n__s(X), tail(X) -> n__tail(X), tail(cons(N, XS)) -> U211(isNatural(N), activate(XS)), take(N, XS) -> U221(isNatural(N), N, XS), take(X1, X2) -> n__take(X1, X2) } DUP: We consider a duplicating system. Trs: { U102(tt()) -> tt(), isLNat(n__natsFrom(V1)) -> U71(isNatural(activate(V1))), isLNat(n__nil()) -> tt(), isLNat(n__afterNth(V1, V2)) -> U41(isNatural(activate(V1)), activate(V2)), isLNat(n__cons(V1, V2)) -> U51(isNatural(activate(V1)), activate(V2)), isLNat(n__fst(V1)) -> U61(isPLNat(activate(V1))), isLNat(n__snd(V1)) -> U81(isPLNat(activate(V1))), isLNat(n__tail(V1)) -> U91(isLNat(activate(V1))), isLNat(n__take(V1, V2)) -> U101(isNatural(activate(V1)), activate(V2)), activate(X) -> X, activate(n__natsFrom(X)) -> natsFrom(activate(X)), activate(n__s(X)) -> s(activate(X)), activate(n__nil()) -> nil(), activate(n__afterNth(X1, X2)) -> afterNth(activate(X1), activate(X2)), activate(n__cons(X1, X2)) -> cons(activate(X1), X2), activate(n__fst(X)) -> fst(activate(X)), activate(n__snd(X)) -> snd(activate(X)), activate(n__tail(X)) -> tail(activate(X)), activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)), activate(n__0()) -> 0(), activate(n__head(X)) -> head(activate(X)), activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2)), activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2)), activate(n__splitAt(X1, X2)) -> splitAt(activate(X1), activate(X2)), U101(tt(), V2) -> U102(isLNat(activate(V2))), U12(tt(), N, XS) -> snd(splitAt(activate(N), activate(XS))), U11(tt(), N, XS) -> U12(isLNat(activate(XS)), activate(N), activate(XS)), U111(tt()) -> tt(), snd(X) -> n__snd(X), snd(pair(X, Y)) -> U181(isLNat(X), Y), splitAt(X1, X2) -> n__splitAt(X1, X2), splitAt(0(), XS) -> U191(isLNat(XS), XS), splitAt(s(N), cons(X, XS)) -> U201(isNatural(N), N, X, activate(XS)), U121(tt()) -> tt(), U132(tt()) -> tt(), U131(tt(), V2) -> U132(isLNat(activate(V2))), U142(tt()) -> tt(), U141(tt(), V2) -> U142(isLNat(activate(V2))), U152(tt()) -> tt(), U151(tt(), V2) -> U152(isLNat(activate(V2))), cons(X1, X2) -> n__cons(X1, X2), U161(tt(), N) -> cons(activate(N), n__natsFrom(n__s(activate(N)))), U172(tt(), N, XS) -> head(afterNth(activate(N), activate(XS))), U171(tt(), N, XS) -> U172(isLNat(activate(XS)), activate(N), activate(XS)), head(X) -> n__head(X), head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)), afterNth(N, XS) -> U11(isNatural(N), N, XS), afterNth(X1, X2) -> n__afterNth(X1, X2), U182(tt(), Y) -> activate(Y), U181(tt(), Y) -> U182(isLNat(activate(Y)), activate(Y)), pair(X1, X2) -> n__pair(X1, X2), nil() -> n__nil(), U191(tt(), XS) -> pair(nil(), activate(XS)), U202(tt(), N, X, XS) -> U203(isLNat(activate(XS)), activate(N), activate(X), activate(XS)), isNatural(n__s(V1)) -> U121(isNatural(activate(V1))), isNatural(n__0()) -> tt(), isNatural(n__head(V1)) -> U111(isLNat(activate(V1))), isNatural(n__sel(V1, V2)) -> U131(isNatural(activate(V1)), activate(V2)), U201(tt(), N, X, XS) -> U202(isNatural(activate(X)), activate(N), activate(X), activate(XS)), U203(tt(), N, X, XS) -> U204(splitAt(activate(N), activate(XS)), activate(X)), U204(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS), U22(tt(), X) -> activate(X), U21(tt(), X, Y) -> U22(isLNat(activate(Y)), activate(X)), U212(tt(), XS) -> activate(XS), U211(tt(), XS) -> U212(isLNat(activate(XS)), activate(XS)), U222(tt(), N, XS) -> fst(splitAt(activate(N), activate(XS))), U221(tt(), N, XS) -> U222(isLNat(activate(XS)), activate(N), activate(XS)), fst(X) -> n__fst(X), fst(pair(X, Y)) -> U21(isLNat(X), X, Y), U32(tt(), N) -> activate(N), U31(tt(), N, XS) -> U32(isLNat(activate(XS)), activate(N)), U42(tt()) -> tt(), U41(tt(), V2) -> U42(isLNat(activate(V2))), U52(tt()) -> tt(), U51(tt(), V2) -> U52(isLNat(activate(V2))), U61(tt()) -> tt(), U71(tt()) -> tt(), U81(tt()) -> tt(), U91(tt()) -> tt(), isPLNat(n__pair(V1, V2)) -> U141(isLNat(activate(V1)), activate(V2)), isPLNat(n__splitAt(V1, V2)) -> U151(isNatural(activate(V1)), activate(V2)), natsFrom(N) -> U161(isNatural(N), N), natsFrom(X) -> n__natsFrom(X), sel(N, XS) -> U171(isNatural(N), N, XS), sel(X1, X2) -> n__sel(X1, X2), 0() -> n__0(), s(X) -> n__s(X), tail(X) -> n__tail(X), tail(cons(N, XS)) -> U211(isNatural(N), activate(XS)), take(N, XS) -> U221(isNatural(N), N, XS), take(X1, X2) -> n__take(X1, X2) } Fail