MAYBE

Problem:
 active(U101(tt(),V2)) -> mark(U102(isLNat(V2)))
 active(U102(tt())) -> mark(tt())
 active(U11(tt(),N,XS)) -> mark(U12(isLNat(XS),N,XS))
 active(U111(tt())) -> mark(tt())
 active(U12(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
 active(U121(tt())) -> mark(tt())
 active(U131(tt(),V2)) -> mark(U132(isLNat(V2)))
 active(U132(tt())) -> mark(tt())
 active(U141(tt(),V2)) -> mark(U142(isLNat(V2)))
 active(U142(tt())) -> mark(tt())
 active(U151(tt(),V2)) -> mark(U152(isLNat(V2)))
 active(U152(tt())) -> mark(tt())
 active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
 active(U171(tt(),N,XS)) -> mark(U172(isLNat(XS),N,XS))
 active(U172(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
 active(U181(tt(),Y)) -> mark(U182(isLNat(Y),Y))
 active(U182(tt(),Y)) -> mark(Y)
 active(U191(tt(),XS)) -> mark(pair(nil(),XS))
 active(U201(tt(),N,X,XS)) -> mark(U202(isNatural(X),N,X,XS))
 active(U202(tt(),N,X,XS)) -> mark(U203(isLNat(XS),N,X,XS))
 active(U203(tt(),N,X,XS)) -> mark(U204(splitAt(N,XS),X))
 active(U204(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
 active(U21(tt(),X,Y)) -> mark(U22(isLNat(Y),X))
 active(U211(tt(),XS)) -> mark(U212(isLNat(XS),XS))
 active(U212(tt(),XS)) -> mark(XS)
 active(U22(tt(),X)) -> mark(X)
 active(U221(tt(),N,XS)) -> mark(U222(isLNat(XS),N,XS))
 active(U222(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
 active(U31(tt(),N,XS)) -> mark(U32(isLNat(XS),N))
 active(U32(tt(),N)) -> mark(N)
 active(U41(tt(),V2)) -> mark(U42(isLNat(V2)))
 active(U42(tt())) -> mark(tt())
 active(U51(tt(),V2)) -> mark(U52(isLNat(V2)))
 active(U52(tt())) -> mark(tt())
 active(U61(tt())) -> mark(tt())
 active(U71(tt())) -> mark(tt())
 active(U81(tt())) -> mark(tt())
 active(U91(tt())) -> mark(tt())
 active(afterNth(N,XS)) -> mark(U11(isNatural(N),N,XS))
 active(fst(pair(X,Y))) -> mark(U21(isLNat(X),X,Y))
 active(head(cons(N,XS))) -> mark(U31(isNatural(N),N,XS))
 active(isLNat(nil())) -> mark(tt())
 active(isLNat(afterNth(V1,V2))) -> mark(U41(isNatural(V1),V2))
 active(isLNat(cons(V1,V2))) -> mark(U51(isNatural(V1),V2))
 active(isLNat(fst(V1))) -> mark(U61(isPLNat(V1)))
 active(isLNat(natsFrom(V1))) -> mark(U71(isNatural(V1)))
 active(isLNat(snd(V1))) -> mark(U81(isPLNat(V1)))
 active(isLNat(tail(V1))) -> mark(U91(isLNat(V1)))
 active(isLNat(take(V1,V2))) -> mark(U101(isNatural(V1),V2))
 active(isNatural(0())) -> mark(tt())
 active(isNatural(head(V1))) -> mark(U111(isLNat(V1)))
 active(isNatural(s(V1))) -> mark(U121(isNatural(V1)))
 active(isNatural(sel(V1,V2))) -> mark(U131(isNatural(V1),V2))
 active(isPLNat(pair(V1,V2))) -> mark(U141(isLNat(V1),V2))
 active(isPLNat(splitAt(V1,V2))) -> mark(U151(isNatural(V1),V2))
 active(natsFrom(N)) -> mark(U161(isNatural(N),N))
 active(sel(N,XS)) -> mark(U171(isNatural(N),N,XS))
 active(snd(pair(X,Y))) -> mark(U181(isLNat(X),Y))
 active(splitAt(0(),XS)) -> mark(U191(isLNat(XS),XS))
 active(splitAt(s(N),cons(X,XS))) -> mark(U201(isNatural(N),N,X,XS))
 active(tail(cons(N,XS))) -> mark(U211(isNatural(N),XS))
 active(take(N,XS)) -> mark(U221(isNatural(N),N,XS))
 mark(U101(X1,X2)) -> active(U101(mark(X1),X2))
 mark(tt()) -> active(tt())
 mark(U102(X)) -> active(U102(mark(X)))
 mark(isLNat(X)) -> active(isLNat(X))
 mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3))
 mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3))
 mark(U111(X)) -> active(U111(mark(X)))
 mark(snd(X)) -> active(snd(mark(X)))
 mark(splitAt(X1,X2)) -> active(splitAt(mark(X1),mark(X2)))
 mark(U121(X)) -> active(U121(mark(X)))
 mark(U131(X1,X2)) -> active(U131(mark(X1),X2))
 mark(U132(X)) -> active(U132(mark(X)))
 mark(U141(X1,X2)) -> active(U141(mark(X1),X2))
 mark(U142(X)) -> active(U142(mark(X)))
 mark(U151(X1,X2)) -> active(U151(mark(X1),X2))
 mark(U152(X)) -> active(U152(mark(X)))
 mark(U161(X1,X2)) -> active(U161(mark(X1),X2))
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(natsFrom(X)) -> active(natsFrom(mark(X)))
 mark(s(X)) -> active(s(mark(X)))
 mark(U171(X1,X2,X3)) -> active(U171(mark(X1),X2,X3))
 mark(U172(X1,X2,X3)) -> active(U172(mark(X1),X2,X3))
 mark(head(X)) -> active(head(mark(X)))
 mark(afterNth(X1,X2)) -> active(afterNth(mark(X1),mark(X2)))
 mark(U181(X1,X2)) -> active(U181(mark(X1),X2))
 mark(U182(X1,X2)) -> active(U182(mark(X1),X2))
 mark(U191(X1,X2)) -> active(U191(mark(X1),X2))
 mark(pair(X1,X2)) -> active(pair(mark(X1),mark(X2)))
 mark(nil()) -> active(nil())
 mark(U201(X1,X2,X3,X4)) -> active(U201(mark(X1),X2,X3,X4))
 mark(U202(X1,X2,X3,X4)) -> active(U202(mark(X1),X2,X3,X4))
 mark(isNatural(X)) -> active(isNatural(X))
 mark(U203(X1,X2,X3,X4)) -> active(U203(mark(X1),X2,X3,X4))
 mark(U204(X1,X2)) -> active(U204(mark(X1),X2))
 mark(U21(X1,X2,X3)) -> active(U21(mark(X1),X2,X3))
 mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
 mark(U211(X1,X2)) -> active(U211(mark(X1),X2))
 mark(U212(X1,X2)) -> active(U212(mark(X1),X2))
 mark(U221(X1,X2,X3)) -> active(U221(mark(X1),X2,X3))
 mark(U222(X1,X2,X3)) -> active(U222(mark(X1),X2,X3))
 mark(fst(X)) -> active(fst(mark(X)))
 mark(U31(X1,X2,X3)) -> active(U31(mark(X1),X2,X3))
 mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
 mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
 mark(U42(X)) -> active(U42(mark(X)))
 mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
 mark(U52(X)) -> active(U52(mark(X)))
 mark(U61(X)) -> active(U61(mark(X)))
 mark(U71(X)) -> active(U71(mark(X)))
 mark(U81(X)) -> active(U81(mark(X)))
 mark(U91(X)) -> active(U91(mark(X)))
 mark(isPLNat(X)) -> active(isPLNat(X))
 mark(tail(X)) -> active(tail(mark(X)))
 mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
 mark(0()) -> active(0())
 mark(sel(X1,X2)) -> active(sel(mark(X1),mark(X2)))
 U101(mark(X1),X2) -> U101(X1,X2)
 U101(X1,mark(X2)) -> U101(X1,X2)
 U101(active(X1),X2) -> U101(X1,X2)
 U101(X1,active(X2)) -> U101(X1,X2)
 U102(mark(X)) -> U102(X)
 U102(active(X)) -> U102(X)
 isLNat(mark(X)) -> isLNat(X)
 isLNat(active(X)) -> isLNat(X)
 U11(mark(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,mark(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,mark(X3)) -> U11(X1,X2,X3)
 U11(active(X1),X2,X3) -> U11(X1,X2,X3)
 U11(X1,active(X2),X3) -> U11(X1,X2,X3)
 U11(X1,X2,active(X3)) -> U11(X1,X2,X3)
 U12(mark(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,mark(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,mark(X3)) -> U12(X1,X2,X3)
 U12(active(X1),X2,X3) -> U12(X1,X2,X3)
 U12(X1,active(X2),X3) -> U12(X1,X2,X3)
 U12(X1,X2,active(X3)) -> U12(X1,X2,X3)
 U111(mark(X)) -> U111(X)
 U111(active(X)) -> U111(X)
 snd(mark(X)) -> snd(X)
 snd(active(X)) -> snd(X)
 splitAt(mark(X1),X2) -> splitAt(X1,X2)
 splitAt(X1,mark(X2)) -> splitAt(X1,X2)
 splitAt(active(X1),X2) -> splitAt(X1,X2)
 splitAt(X1,active(X2)) -> splitAt(X1,X2)
 U121(mark(X)) -> U121(X)
 U121(active(X)) -> U121(X)
 U131(mark(X1),X2) -> U131(X1,X2)
 U131(X1,mark(X2)) -> U131(X1,X2)
 U131(active(X1),X2) -> U131(X1,X2)
 U131(X1,active(X2)) -> U131(X1,X2)
 U132(mark(X)) -> U132(X)
 U132(active(X)) -> U132(X)
 U141(mark(X1),X2) -> U141(X1,X2)
 U141(X1,mark(X2)) -> U141(X1,X2)
 U141(active(X1),X2) -> U141(X1,X2)
 U141(X1,active(X2)) -> U141(X1,X2)
 U142(mark(X)) -> U142(X)
 U142(active(X)) -> U142(X)
 U151(mark(X1),X2) -> U151(X1,X2)
 U151(X1,mark(X2)) -> U151(X1,X2)
 U151(active(X1),X2) -> U151(X1,X2)
 U151(X1,active(X2)) -> U151(X1,X2)
 U152(mark(X)) -> U152(X)
 U152(active(X)) -> U152(X)
 U161(mark(X1),X2) -> U161(X1,X2)
 U161(X1,mark(X2)) -> U161(X1,X2)
 U161(active(X1),X2) -> U161(X1,X2)
 U161(X1,active(X2)) -> U161(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 cons(active(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 natsFrom(mark(X)) -> natsFrom(X)
 natsFrom(active(X)) -> natsFrom(X)
 s(mark(X)) -> s(X)
 s(active(X)) -> s(X)
 U171(mark(X1),X2,X3) -> U171(X1,X2,X3)
 U171(X1,mark(X2),X3) -> U171(X1,X2,X3)
 U171(X1,X2,mark(X3)) -> U171(X1,X2,X3)
 U171(active(X1),X2,X3) -> U171(X1,X2,X3)
 U171(X1,active(X2),X3) -> U171(X1,X2,X3)
 U171(X1,X2,active(X3)) -> U171(X1,X2,X3)
 U172(mark(X1),X2,X3) -> U172(X1,X2,X3)
 U172(X1,mark(X2),X3) -> U172(X1,X2,X3)
 U172(X1,X2,mark(X3)) -> U172(X1,X2,X3)
 U172(active(X1),X2,X3) -> U172(X1,X2,X3)
 U172(X1,active(X2),X3) -> U172(X1,X2,X3)
 U172(X1,X2,active(X3)) -> U172(X1,X2,X3)
 head(mark(X)) -> head(X)
 head(active(X)) -> head(X)
 afterNth(mark(X1),X2) -> afterNth(X1,X2)
 afterNth(X1,mark(X2)) -> afterNth(X1,X2)
 afterNth(active(X1),X2) -> afterNth(X1,X2)
 afterNth(X1,active(X2)) -> afterNth(X1,X2)
 U181(mark(X1),X2) -> U181(X1,X2)
 U181(X1,mark(X2)) -> U181(X1,X2)
 U181(active(X1),X2) -> U181(X1,X2)
 U181(X1,active(X2)) -> U181(X1,X2)
 U182(mark(X1),X2) -> U182(X1,X2)
 U182(X1,mark(X2)) -> U182(X1,X2)
 U182(active(X1),X2) -> U182(X1,X2)
 U182(X1,active(X2)) -> U182(X1,X2)
 U191(mark(X1),X2) -> U191(X1,X2)
 U191(X1,mark(X2)) -> U191(X1,X2)
 U191(active(X1),X2) -> U191(X1,X2)
 U191(X1,active(X2)) -> U191(X1,X2)
 pair(mark(X1),X2) -> pair(X1,X2)
 pair(X1,mark(X2)) -> pair(X1,X2)
 pair(active(X1),X2) -> pair(X1,X2)
 pair(X1,active(X2)) -> pair(X1,X2)
 U201(mark(X1),X2,X3,X4) -> U201(X1,X2,X3,X4)
 U201(X1,mark(X2),X3,X4) -> U201(X1,X2,X3,X4)
 U201(X1,X2,mark(X3),X4) -> U201(X1,X2,X3,X4)
 U201(X1,X2,X3,mark(X4)) -> U201(X1,X2,X3,X4)
 U201(active(X1),X2,X3,X4) -> U201(X1,X2,X3,X4)
 U201(X1,active(X2),X3,X4) -> U201(X1,X2,X3,X4)
 U201(X1,X2,active(X3),X4) -> U201(X1,X2,X3,X4)
 U201(X1,X2,X3,active(X4)) -> U201(X1,X2,X3,X4)
 U202(mark(X1),X2,X3,X4) -> U202(X1,X2,X3,X4)
 U202(X1,mark(X2),X3,X4) -> U202(X1,X2,X3,X4)
 U202(X1,X2,mark(X3),X4) -> U202(X1,X2,X3,X4)
 U202(X1,X2,X3,mark(X4)) -> U202(X1,X2,X3,X4)
 U202(active(X1),X2,X3,X4) -> U202(X1,X2,X3,X4)
 U202(X1,active(X2),X3,X4) -> U202(X1,X2,X3,X4)
 U202(X1,X2,active(X3),X4) -> U202(X1,X2,X3,X4)
 U202(X1,X2,X3,active(X4)) -> U202(X1,X2,X3,X4)
 isNatural(mark(X)) -> isNatural(X)
 isNatural(active(X)) -> isNatural(X)
 U203(mark(X1),X2,X3,X4) -> U203(X1,X2,X3,X4)
 U203(X1,mark(X2),X3,X4) -> U203(X1,X2,X3,X4)
 U203(X1,X2,mark(X3),X4) -> U203(X1,X2,X3,X4)
 U203(X1,X2,X3,mark(X4)) -> U203(X1,X2,X3,X4)
 U203(active(X1),X2,X3,X4) -> U203(X1,X2,X3,X4)
 U203(X1,active(X2),X3,X4) -> U203(X1,X2,X3,X4)
 U203(X1,X2,active(X3),X4) -> U203(X1,X2,X3,X4)
 U203(X1,X2,X3,active(X4)) -> U203(X1,X2,X3,X4)
 U204(mark(X1),X2) -> U204(X1,X2)
 U204(X1,mark(X2)) -> U204(X1,X2)
 U204(active(X1),X2) -> U204(X1,X2)
 U204(X1,active(X2)) -> U204(X1,X2)
 U21(mark(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,mark(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,mark(X3)) -> U21(X1,X2,X3)
 U21(active(X1),X2,X3) -> U21(X1,X2,X3)
 U21(X1,active(X2),X3) -> U21(X1,X2,X3)
 U21(X1,X2,active(X3)) -> U21(X1,X2,X3)
 U22(mark(X1),X2) -> U22(X1,X2)
 U22(X1,mark(X2)) -> U22(X1,X2)
 U22(active(X1),X2) -> U22(X1,X2)
 U22(X1,active(X2)) -> U22(X1,X2)
 U211(mark(X1),X2) -> U211(X1,X2)
 U211(X1,mark(X2)) -> U211(X1,X2)
 U211(active(X1),X2) -> U211(X1,X2)
 U211(X1,active(X2)) -> U211(X1,X2)
 U212(mark(X1),X2) -> U212(X1,X2)
 U212(X1,mark(X2)) -> U212(X1,X2)
 U212(active(X1),X2) -> U212(X1,X2)
 U212(X1,active(X2)) -> U212(X1,X2)
 U221(mark(X1),X2,X3) -> U221(X1,X2,X3)
 U221(X1,mark(X2),X3) -> U221(X1,X2,X3)
 U221(X1,X2,mark(X3)) -> U221(X1,X2,X3)
 U221(active(X1),X2,X3) -> U221(X1,X2,X3)
 U221(X1,active(X2),X3) -> U221(X1,X2,X3)
 U221(X1,X2,active(X3)) -> U221(X1,X2,X3)
 U222(mark(X1),X2,X3) -> U222(X1,X2,X3)
 U222(X1,mark(X2),X3) -> U222(X1,X2,X3)
 U222(X1,X2,mark(X3)) -> U222(X1,X2,X3)
 U222(active(X1),X2,X3) -> U222(X1,X2,X3)
 U222(X1,active(X2),X3) -> U222(X1,X2,X3)
 U222(X1,X2,active(X3)) -> U222(X1,X2,X3)
 fst(mark(X)) -> fst(X)
 fst(active(X)) -> fst(X)
 U31(mark(X1),X2,X3) -> U31(X1,X2,X3)
 U31(X1,mark(X2),X3) -> U31(X1,X2,X3)
 U31(X1,X2,mark(X3)) -> U31(X1,X2,X3)
 U31(active(X1),X2,X3) -> U31(X1,X2,X3)
 U31(X1,active(X2),X3) -> U31(X1,X2,X3)
 U31(X1,X2,active(X3)) -> U31(X1,X2,X3)
 U32(mark(X1),X2) -> U32(X1,X2)
 U32(X1,mark(X2)) -> U32(X1,X2)
 U32(active(X1),X2) -> U32(X1,X2)
 U32(X1,active(X2)) -> U32(X1,X2)
 U41(mark(X1),X2) -> U41(X1,X2)
 U41(X1,mark(X2)) -> U41(X1,X2)
 U41(active(X1),X2) -> U41(X1,X2)
 U41(X1,active(X2)) -> U41(X1,X2)
 U42(mark(X)) -> U42(X)
 U42(active(X)) -> U42(X)
 U51(mark(X1),X2) -> U51(X1,X2)
 U51(X1,mark(X2)) -> U51(X1,X2)
 U51(active(X1),X2) -> U51(X1,X2)
 U51(X1,active(X2)) -> U51(X1,X2)
 U52(mark(X)) -> U52(X)
 U52(active(X)) -> U52(X)
 U61(mark(X)) -> U61(X)
 U61(active(X)) -> U61(X)
 U71(mark(X)) -> U71(X)
 U71(active(X)) -> U71(X)
 U81(mark(X)) -> U81(X)
 U81(active(X)) -> U81(X)
 U91(mark(X)) -> U91(X)
 U91(active(X)) -> U91(X)
 isPLNat(mark(X)) -> isPLNat(X)
 isPLNat(active(X)) -> isPLNat(X)
 tail(mark(X)) -> tail(X)
 tail(active(X)) -> tail(X)
 take(mark(X1),X2) -> take(X1,X2)
 take(X1,mark(X2)) -> take(X1,X2)
 take(active(X1),X2) -> take(X1,X2)
 take(X1,active(X2)) -> take(X1,X2)
 sel(mark(X1),X2) -> sel(X1,X2)
 sel(X1,mark(X2)) -> sel(X1,X2)
 sel(active(X1),X2) -> sel(X1,X2)
 sel(X1,active(X2)) -> sel(X1,X2)

Proof:
 Open