MAYBE

Problem:
 active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
 active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
 active(U103(tt())) -> mark(tt())
 active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
 active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
 active(U112(tt())) -> mark(tt())
 active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
 active(U122(tt())) -> mark(tt())
 active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
 active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
 active(U133(tt())) -> mark(tt())
 active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
 active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
 active(U143(tt())) -> mark(tt())
 active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
 active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
 active(U153(tt())) -> mark(tt())
 active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
 active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
 active(U181(tt(),Y)) -> mark(Y)
 active(U191(tt(),XS)) -> mark(pair(nil(),XS))
 active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
 active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
 active(U21(tt(),X)) -> mark(X)
 active(U211(tt(),XS)) -> mark(XS)
 active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
 active(U31(tt(),N)) -> mark(N)
 active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
 active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
 active(U43(tt())) -> mark(tt())
 active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
 active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
 active(U53(tt())) -> mark(tt())
 active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
 active(U62(tt())) -> mark(tt())
 active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
 active(U72(tt())) -> mark(tt())
 active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
 active(U82(tt())) -> mark(tt())
 active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
 active(U92(tt())) -> mark(tt())
 active(afterNth(N,XS)) ->
 mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
 active(and(tt(),X)) -> mark(X)
 active(fst(pair(X,Y))) ->
 mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
 active(head(cons(N,XS))) ->
 mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
 active(isLNat(nil())) -> mark(tt())
 active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
 active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
 active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
 active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
 active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
 active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
 active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
 active(isLNatKind(nil())) -> mark(tt())
 active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
 active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
 active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
 active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
 active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
 active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
 active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
 active(isNatural(0())) -> mark(tt())
 active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
 active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
 active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
 active(isNaturalKind(0())) -> mark(tt())
 active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
 active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
 active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
 active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
 active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
 active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
 active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
 active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
 active(sel(N,XS)) ->
 mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
 active(snd(pair(X,Y))) ->
 mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
 active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
 active(splitAt(s(N),cons(X,XS))) ->
 mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                      and(isLNat(XS),isLNatKind(XS)))),
           N,X,XS))
 active(tail(cons(N,XS))) ->
 mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
 active(take(N,XS)) ->
 mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
 active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
 active(U102(X1,X2)) -> U102(active(X1),X2)
 active(U103(X)) -> U103(active(X))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(snd(X)) -> snd(active(X))
 active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
 active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
 active(U111(X1,X2)) -> U111(active(X1),X2)
 active(U112(X)) -> U112(active(X))
 active(U121(X1,X2)) -> U121(active(X1),X2)
 active(U122(X)) -> U122(active(X))
 active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
 active(U132(X1,X2)) -> U132(active(X1),X2)
 active(U133(X)) -> U133(active(X))
 active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
 active(U142(X1,X2)) -> U142(active(X1),X2)
 active(U143(X)) -> U143(active(X))
 active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
 active(U152(X1,X2)) -> U152(active(X1),X2)
 active(U153(X)) -> U153(active(X))
 active(U161(X1,X2)) -> U161(active(X1),X2)
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(natsFrom(X)) -> natsFrom(active(X))
 active(s(X)) -> s(active(X))
 active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
 active(head(X)) -> head(active(X))
 active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
 active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
 active(U181(X1,X2)) -> U181(active(X1),X2)
 active(U191(X1,X2)) -> U191(active(X1),X2)
 active(pair(X1,X2)) -> pair(active(X1),X2)
 active(pair(X1,X2)) -> pair(X1,active(X2))
 active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
 active(U202(X1,X2)) -> U202(active(X1),X2)
 active(U21(X1,X2)) -> U21(active(X1),X2)
 active(U211(X1,X2)) -> U211(active(X1),X2)
 active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
 active(fst(X)) -> fst(active(X))
 active(U31(X1,X2)) -> U31(active(X1),X2)
 active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
 active(U42(X1,X2)) -> U42(active(X1),X2)
 active(U43(X)) -> U43(active(X))
 active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
 active(U52(X1,X2)) -> U52(active(X1),X2)
 active(U53(X)) -> U53(active(X))
 active(U61(X1,X2)) -> U61(active(X1),X2)
 active(U62(X)) -> U62(active(X))
 active(U71(X1,X2)) -> U71(active(X1),X2)
 active(U72(X)) -> U72(active(X))
 active(U81(X1,X2)) -> U81(active(X1),X2)
 active(U82(X)) -> U82(active(X))
 active(U91(X1,X2)) -> U91(active(X1),X2)
 active(U92(X)) -> U92(active(X))
 active(and(X1,X2)) -> and(active(X1),X2)
 active(tail(X)) -> tail(active(X))
 active(take(X1,X2)) -> take(active(X1),X2)
 active(take(X1,X2)) -> take(X1,active(X2))
 active(sel(X1,X2)) -> sel(active(X1),X2)
 active(sel(X1,X2)) -> sel(X1,active(X2))
 U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
 U102(mark(X1),X2) -> mark(U102(X1,X2))
 U103(mark(X)) -> mark(U103(X))
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 snd(mark(X)) -> mark(snd(X))
 splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
 splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
 U111(mark(X1),X2) -> mark(U111(X1,X2))
 U112(mark(X)) -> mark(U112(X))
 U121(mark(X1),X2) -> mark(U121(X1,X2))
 U122(mark(X)) -> mark(U122(X))
 U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
 U132(mark(X1),X2) -> mark(U132(X1,X2))
 U133(mark(X)) -> mark(U133(X))
 U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
 U142(mark(X1),X2) -> mark(U142(X1,X2))
 U143(mark(X)) -> mark(U143(X))
 U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
 U152(mark(X1),X2) -> mark(U152(X1,X2))
 U153(mark(X)) -> mark(U153(X))
 U161(mark(X1),X2) -> mark(U161(X1,X2))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 natsFrom(mark(X)) -> mark(natsFrom(X))
 s(mark(X)) -> mark(s(X))
 U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
 head(mark(X)) -> mark(head(X))
 afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
 afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
 U181(mark(X1),X2) -> mark(U181(X1,X2))
 U191(mark(X1),X2) -> mark(U191(X1,X2))
 pair(mark(X1),X2) -> mark(pair(X1,X2))
 pair(X1,mark(X2)) -> mark(pair(X1,X2))
 U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
 U202(mark(X1),X2) -> mark(U202(X1,X2))
 U21(mark(X1),X2) -> mark(U21(X1,X2))
 U211(mark(X1),X2) -> mark(U211(X1,X2))
 U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
 fst(mark(X)) -> mark(fst(X))
 U31(mark(X1),X2) -> mark(U31(X1,X2))
 U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
 U42(mark(X1),X2) -> mark(U42(X1,X2))
 U43(mark(X)) -> mark(U43(X))
 U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
 U52(mark(X1),X2) -> mark(U52(X1,X2))
 U53(mark(X)) -> mark(U53(X))
 U61(mark(X1),X2) -> mark(U61(X1,X2))
 U62(mark(X)) -> mark(U62(X))
 U71(mark(X1),X2) -> mark(U71(X1,X2))
 U72(mark(X)) -> mark(U72(X))
 U81(mark(X1),X2) -> mark(U81(X1,X2))
 U82(mark(X)) -> mark(U82(X))
 U91(mark(X1),X2) -> mark(U91(X1,X2))
 U92(mark(X)) -> mark(U92(X))
 and(mark(X1),X2) -> mark(and(X1,X2))
 tail(mark(X)) -> mark(tail(X))
 take(mark(X1),X2) -> mark(take(X1,X2))
 take(X1,mark(X2)) -> mark(take(X1,X2))
 sel(mark(X1),X2) -> mark(sel(X1,X2))
 sel(X1,mark(X2)) -> mark(sel(X1,X2))
 proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
 proper(tt()) -> ok(tt())
 proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
 proper(isNatural(X)) -> isNatural(proper(X))
 proper(U103(X)) -> U103(proper(X))
 proper(isLNat(X)) -> isLNat(proper(X))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(snd(X)) -> snd(proper(X))
 proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
 proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
 proper(U112(X)) -> U112(proper(X))
 proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
 proper(U122(X)) -> U122(proper(X))
 proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
 proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
 proper(U133(X)) -> U133(proper(X))
 proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
 proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
 proper(U143(X)) -> U143(proper(X))
 proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
 proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
 proper(U153(X)) -> U153(proper(X))
 proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(natsFrom(X)) -> natsFrom(proper(X))
 proper(s(X)) -> s(proper(X))
 proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
 proper(head(X)) -> head(proper(X))
 proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
 proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
 proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
 proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
 proper(nil()) -> ok(nil())
 proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
 proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
 proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
 proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
 proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
 proper(fst(X)) -> fst(proper(X))
 proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
 proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
 proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
 proper(U43(X)) -> U43(proper(X))
 proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
 proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
 proper(U53(X)) -> U53(proper(X))
 proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
 proper(U62(X)) -> U62(proper(X))
 proper(isPLNat(X)) -> isPLNat(proper(X))
 proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
 proper(U72(X)) -> U72(proper(X))
 proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
 proper(U82(X)) -> U82(proper(X))
 proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
 proper(U92(X)) -> U92(proper(X))
 proper(and(X1,X2)) -> and(proper(X1),proper(X2))
 proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
 proper(isLNatKind(X)) -> isLNatKind(proper(X))
 proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
 proper(tail(X)) -> tail(proper(X))
 proper(take(X1,X2)) -> take(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
 U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
 U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
 isNatural(ok(X)) -> ok(isNatural(X))
 U103(ok(X)) -> ok(U103(X))
 isLNat(ok(X)) -> ok(isLNat(X))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 snd(ok(X)) -> ok(snd(X))
 splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
 U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
 U112(ok(X)) -> ok(U112(X))
 U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
 U122(ok(X)) -> ok(U122(X))
 U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
 U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
 U133(ok(X)) -> ok(U133(X))
 U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
 U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
 U143(ok(X)) -> ok(U143(X))
 U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
 U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
 U153(ok(X)) -> ok(U153(X))
 U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 natsFrom(ok(X)) -> ok(natsFrom(X))
 s(ok(X)) -> ok(s(X))
 U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
 head(ok(X)) -> ok(head(X))
 afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
 U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
 U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
 pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
 U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
 U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
 U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
 U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
 U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
 fst(ok(X)) -> ok(fst(X))
 U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
 U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
 U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
 U43(ok(X)) -> ok(U43(X))
 U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
 U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
 U53(ok(X)) -> ok(U53(X))
 U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
 U62(ok(X)) -> ok(U62(X))
 isPLNat(ok(X)) -> ok(isPLNat(X))
 U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
 U72(ok(X)) -> ok(U72(X))
 U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
 U82(ok(X)) -> ok(U82(X))
 U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
 U92(ok(X)) -> ok(U92(X))
 and(ok(X1),ok(X2)) -> ok(and(X1,X2))
 isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
 isLNatKind(ok(X)) -> ok(isLNatKind(X))
 isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
 tail(ok(X)) -> ok(tail(X))
 take(ok(X1),ok(X2)) -> ok(take(X1,X2))
 sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(U101(tt(),V1,V2)) -> isNatural#(V1)
   active#(U101(tt(),V1,V2)) -> U102#(isNatural(V1),V2)
   active#(U102(tt(),V2)) -> isLNat#(V2)
   active#(U102(tt(),V2)) -> U103#(isLNat(V2))
   active#(U11(tt(),N,XS)) -> splitAt#(N,XS)
   active#(U11(tt(),N,XS)) -> snd#(splitAt(N,XS))
   active#(U111(tt(),V1)) -> isLNat#(V1)
   active#(U111(tt(),V1)) -> U112#(isLNat(V1))
   active#(U121(tt(),V1)) -> isNatural#(V1)
   active#(U121(tt(),V1)) -> U122#(isNatural(V1))
   active#(U131(tt(),V1,V2)) -> isNatural#(V1)
   active#(U131(tt(),V1,V2)) -> U132#(isNatural(V1),V2)
   active#(U132(tt(),V2)) -> isLNat#(V2)
   active#(U132(tt(),V2)) -> U133#(isLNat(V2))
   active#(U141(tt(),V1,V2)) -> isLNat#(V1)
   active#(U141(tt(),V1,V2)) -> U142#(isLNat(V1),V2)
   active#(U142(tt(),V2)) -> isLNat#(V2)
   active#(U142(tt(),V2)) -> U143#(isLNat(V2))
   active#(U151(tt(),V1,V2)) -> isNatural#(V1)
   active#(U151(tt(),V1,V2)) -> U152#(isNatural(V1),V2)
   active#(U152(tt(),V2)) -> isLNat#(V2)
   active#(U152(tt(),V2)) -> U153#(isLNat(V2))
   active#(U161(tt(),N)) -> s#(N)
   active#(U161(tt(),N)) -> natsFrom#(s(N))
   active#(U161(tt(),N)) -> cons#(N,natsFrom(s(N)))
   active#(U171(tt(),N,XS)) -> afterNth#(N,XS)
   active#(U171(tt(),N,XS)) -> head#(afterNth(N,XS))
   active#(U191(tt(),XS)) -> pair#(nil(),XS)
   active#(U201(tt(),N,X,XS)) -> splitAt#(N,XS)
   active#(U201(tt(),N,X,XS)) -> U202#(splitAt(N,XS),X)
   active#(U202(pair(YS,ZS),X)) -> cons#(X,YS)
   active#(U202(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS)
   active#(U221(tt(),N,XS)) -> splitAt#(N,XS)
   active#(U221(tt(),N,XS)) -> fst#(splitAt(N,XS))
   active#(U41(tt(),V1,V2)) -> isNatural#(V1)
   active#(U41(tt(),V1,V2)) -> U42#(isNatural(V1),V2)
   active#(U42(tt(),V2)) -> isLNat#(V2)
   active#(U42(tt(),V2)) -> U43#(isLNat(V2))
   active#(U51(tt(),V1,V2)) -> isNatural#(V1)
   active#(U51(tt(),V1,V2)) -> U52#(isNatural(V1),V2)
   active#(U52(tt(),V2)) -> isLNat#(V2)
   active#(U52(tt(),V2)) -> U53#(isLNat(V2))
   active#(U61(tt(),V1)) -> isPLNat#(V1)
   active#(U61(tt(),V1)) -> U62#(isPLNat(V1))
   active#(U71(tt(),V1)) -> isNatural#(V1)
   active#(U71(tt(),V1)) -> U72#(isNatural(V1))
   active#(U81(tt(),V1)) -> isPLNat#(V1)
   active#(U81(tt(),V1)) -> U82#(isPLNat(V1))
   active#(U91(tt(),V1)) -> isLNat#(V1)
   active#(U91(tt(),V1)) -> U92#(isLNat(V1))
   active#(afterNth(N,XS)) -> isLNatKind#(XS)
   active#(afterNth(N,XS)) -> isLNat#(XS)
   active#(afterNth(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
   active#(afterNth(N,XS)) -> isNaturalKind#(N)
   active#(afterNth(N,XS)) -> isNatural#(N)
   active#(afterNth(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
   active#(afterNth(N,XS)) ->
   and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
   active#(afterNth(N,XS)) ->
   U11#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
   active#(fst(pair(X,Y))) -> isLNatKind#(Y)
   active#(fst(pair(X,Y))) -> isLNat#(Y)
   active#(fst(pair(X,Y))) -> and#(isLNat(Y),isLNatKind(Y))
   active#(fst(pair(X,Y))) -> isLNatKind#(X)
   active#(fst(pair(X,Y))) -> isLNat#(X)
   active#(fst(pair(X,Y))) -> and#(isLNat(X),isLNatKind(X))
   active#(fst(pair(X,Y))) -> and#(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y)))
   active#(fst(pair(X,Y))) ->
   U21#(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X)
   active#(head(cons(N,XS))) -> isLNatKind#(XS)
   active#(head(cons(N,XS))) -> isLNat#(XS)
   active#(head(cons(N,XS))) -> and#(isLNat(XS),isLNatKind(XS))
   active#(head(cons(N,XS))) -> isNaturalKind#(N)
   active#(head(cons(N,XS))) -> isNatural#(N)
   active#(head(cons(N,XS))) -> and#(isNatural(N),isNaturalKind(N))
   active#(head(cons(N,XS))) ->
   and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
   active#(head(cons(N,XS))) ->
   U31#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N)
   active#(isLNat(afterNth(V1,V2))) -> isLNatKind#(V2)
   active#(isLNat(afterNth(V1,V2))) -> isNaturalKind#(V1)
   active#(isLNat(afterNth(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isLNat(afterNth(V1,V2))) -> U41#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
   active#(isLNat(cons(V1,V2))) -> isLNatKind#(V2)
   active#(isLNat(cons(V1,V2))) -> isNaturalKind#(V1)
   active#(isLNat(cons(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isLNat(cons(V1,V2))) -> U51#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
   active#(isLNat(fst(V1))) -> isPLNatKind#(V1)
   active#(isLNat(fst(V1))) -> U61#(isPLNatKind(V1),V1)
   active#(isLNat(natsFrom(V1))) -> isNaturalKind#(V1)
   active#(isLNat(natsFrom(V1))) -> U71#(isNaturalKind(V1),V1)
   active#(isLNat(snd(V1))) -> isPLNatKind#(V1)
   active#(isLNat(snd(V1))) -> U81#(isPLNatKind(V1),V1)
   active#(isLNat(tail(V1))) -> isLNatKind#(V1)
   active#(isLNat(tail(V1))) -> U91#(isLNatKind(V1),V1)
   active#(isLNat(take(V1,V2))) -> isLNatKind#(V2)
   active#(isLNat(take(V1,V2))) -> isNaturalKind#(V1)
   active#(isLNat(take(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isLNat(take(V1,V2))) -> U101#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
   active#(isLNatKind(afterNth(V1,V2))) -> isLNatKind#(V2)
   active#(isLNatKind(afterNth(V1,V2))) -> isNaturalKind#(V1)
   active#(isLNatKind(afterNth(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isLNatKind(cons(V1,V2))) -> isLNatKind#(V2)
   active#(isLNatKind(cons(V1,V2))) -> isNaturalKind#(V1)
   active#(isLNatKind(cons(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isLNatKind(fst(V1))) -> isPLNatKind#(V1)
   active#(isLNatKind(natsFrom(V1))) -> isNaturalKind#(V1)
   active#(isLNatKind(snd(V1))) -> isPLNatKind#(V1)
   active#(isLNatKind(tail(V1))) -> isLNatKind#(V1)
   active#(isLNatKind(take(V1,V2))) -> isLNatKind#(V2)
   active#(isLNatKind(take(V1,V2))) -> isNaturalKind#(V1)
   active#(isLNatKind(take(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isNatural(head(V1))) -> isLNatKind#(V1)
   active#(isNatural(head(V1))) -> U111#(isLNatKind(V1),V1)
   active#(isNatural(s(V1))) -> isNaturalKind#(V1)
   active#(isNatural(s(V1))) -> U121#(isNaturalKind(V1),V1)
   active#(isNatural(sel(V1,V2))) -> isLNatKind#(V2)
   active#(isNatural(sel(V1,V2))) -> isNaturalKind#(V1)
   active#(isNatural(sel(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isNatural(sel(V1,V2))) -> U131#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
   active#(isNaturalKind(head(V1))) -> isLNatKind#(V1)
   active#(isNaturalKind(s(V1))) -> isNaturalKind#(V1)
   active#(isNaturalKind(sel(V1,V2))) -> isLNatKind#(V2)
   active#(isNaturalKind(sel(V1,V2))) -> isNaturalKind#(V1)
   active#(isNaturalKind(sel(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isPLNat(pair(V1,V2))) -> isLNatKind#(V2)
   active#(isPLNat(pair(V1,V2))) -> isLNatKind#(V1)
   active#(isPLNat(pair(V1,V2))) -> and#(isLNatKind(V1),isLNatKind(V2))
   active#(isPLNat(pair(V1,V2))) -> U141#(and(isLNatKind(V1),isLNatKind(V2)),V1,V2)
   active#(isPLNat(splitAt(V1,V2))) -> isLNatKind#(V2)
   active#(isPLNat(splitAt(V1,V2))) -> isNaturalKind#(V1)
   active#(isPLNat(splitAt(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(isPLNat(splitAt(V1,V2))) -> U151#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
   active#(isPLNatKind(pair(V1,V2))) -> isLNatKind#(V2)
   active#(isPLNatKind(pair(V1,V2))) -> isLNatKind#(V1)
   active#(isPLNatKind(pair(V1,V2))) -> and#(isLNatKind(V1),isLNatKind(V2))
   active#(isPLNatKind(splitAt(V1,V2))) -> isLNatKind#(V2)
   active#(isPLNatKind(splitAt(V1,V2))) -> isNaturalKind#(V1)
   active#(isPLNatKind(splitAt(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
   active#(natsFrom(N)) -> isNaturalKind#(N)
   active#(natsFrom(N)) -> isNatural#(N)
   active#(natsFrom(N)) -> and#(isNatural(N),isNaturalKind(N))
   active#(natsFrom(N)) -> U161#(and(isNatural(N),isNaturalKind(N)),N)
   active#(sel(N,XS)) -> isLNatKind#(XS)
   active#(sel(N,XS)) -> isLNat#(XS)
   active#(sel(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
   active#(sel(N,XS)) -> isNaturalKind#(N)
   active#(sel(N,XS)) -> isNatural#(N)
   active#(sel(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
   active#(sel(N,XS)) -> and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
   active#(sel(N,XS)) ->
   U171#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
   active#(snd(pair(X,Y))) -> isLNatKind#(Y)
   active#(snd(pair(X,Y))) -> isLNat#(Y)
   active#(snd(pair(X,Y))) -> and#(isLNat(Y),isLNatKind(Y))
   active#(snd(pair(X,Y))) -> isLNatKind#(X)
   active#(snd(pair(X,Y))) -> isLNat#(X)
   active#(snd(pair(X,Y))) -> and#(isLNat(X),isLNatKind(X))
   active#(snd(pair(X,Y))) -> and#(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y)))
   active#(snd(pair(X,Y))) ->
   U181#(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y)
   active#(splitAt(0(),XS)) -> isLNatKind#(XS)
   active#(splitAt(0(),XS)) -> isLNat#(XS)
   active#(splitAt(0(),XS)) -> and#(isLNat(XS),isLNatKind(XS))
   active#(splitAt(0(),XS)) -> U191#(and(isLNat(XS),isLNatKind(XS)),XS)
   active#(splitAt(s(N),cons(X,XS))) -> isLNatKind#(XS)
   active#(splitAt(s(N),cons(X,XS))) -> isLNat#(XS)
   active#(splitAt(s(N),cons(X,XS))) -> and#(isLNat(XS),isLNatKind(XS))
   active#(splitAt(s(N),cons(X,XS))) -> isNaturalKind#(X)
   active#(splitAt(s(N),cons(X,XS))) -> isNatural#(X)
   active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(X),isNaturalKind(X))
   active#(splitAt(s(N),cons(X,XS))) ->
   and#(and(isNatural(X),isNaturalKind(X)),and(isLNat(XS),isLNatKind(XS)))
   active#(splitAt(s(N),cons(X,XS))) -> isNaturalKind#(N)
   active#(splitAt(s(N),cons(X,XS))) -> isNatural#(N)
   active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(N),isNaturalKind(N))
   active#(splitAt(s(N),cons(X,XS))) ->
   and#(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                               and(isLNat(XS),isLNatKind(XS))))
   active#(splitAt(s(N),cons(X,XS))) ->
   U201#(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                    and(isLNat(XS),isLNatKind(XS)))),
         N,X,XS)
   active#(tail(cons(N,XS))) -> isLNatKind#(XS)
   active#(tail(cons(N,XS))) -> isLNat#(XS)
   active#(tail(cons(N,XS))) -> and#(isLNat(XS),isLNatKind(XS))
   active#(tail(cons(N,XS))) -> isNaturalKind#(N)
   active#(tail(cons(N,XS))) -> isNatural#(N)
   active#(tail(cons(N,XS))) -> and#(isNatural(N),isNaturalKind(N))
   active#(tail(cons(N,XS))) ->
   and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
   active#(tail(cons(N,XS))) ->
   U211#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS)
   active#(take(N,XS)) -> isLNatKind#(XS)
   active#(take(N,XS)) -> isLNat#(XS)
   active#(take(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
   active#(take(N,XS)) -> isNaturalKind#(N)
   active#(take(N,XS)) -> isNatural#(N)
   active#(take(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
   active#(take(N,XS)) -> and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
   active#(take(N,XS)) ->
   U221#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
   active#(U101(X1,X2,X3)) -> active#(X1)
   active#(U101(X1,X2,X3)) -> U101#(active(X1),X2,X3)
   active#(U102(X1,X2)) -> active#(X1)
   active#(U102(X1,X2)) -> U102#(active(X1),X2)
   active#(U103(X)) -> active#(X)
   active#(U103(X)) -> U103#(active(X))
   active#(U11(X1,X2,X3)) -> active#(X1)
   active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
   active#(snd(X)) -> active#(X)
   active#(snd(X)) -> snd#(active(X))
   active#(splitAt(X1,X2)) -> active#(X1)
   active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2)
   active#(splitAt(X1,X2)) -> active#(X2)
   active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2))
   active#(U111(X1,X2)) -> active#(X1)
   active#(U111(X1,X2)) -> U111#(active(X1),X2)
   active#(U112(X)) -> active#(X)
   active#(U112(X)) -> U112#(active(X))
   active#(U121(X1,X2)) -> active#(X1)
   active#(U121(X1,X2)) -> U121#(active(X1),X2)
   active#(U122(X)) -> active#(X)
   active#(U122(X)) -> U122#(active(X))
   active#(U131(X1,X2,X3)) -> active#(X1)
   active#(U131(X1,X2,X3)) -> U131#(active(X1),X2,X3)
   active#(U132(X1,X2)) -> active#(X1)
   active#(U132(X1,X2)) -> U132#(active(X1),X2)
   active#(U133(X)) -> active#(X)
   active#(U133(X)) -> U133#(active(X))
   active#(U141(X1,X2,X3)) -> active#(X1)
   active#(U141(X1,X2,X3)) -> U141#(active(X1),X2,X3)
   active#(U142(X1,X2)) -> active#(X1)
   active#(U142(X1,X2)) -> U142#(active(X1),X2)
   active#(U143(X)) -> active#(X)
   active#(U143(X)) -> U143#(active(X))
   active#(U151(X1,X2,X3)) -> active#(X1)
   active#(U151(X1,X2,X3)) -> U151#(active(X1),X2,X3)
   active#(U152(X1,X2)) -> active#(X1)
   active#(U152(X1,X2)) -> U152#(active(X1),X2)
   active#(U153(X)) -> active#(X)
   active#(U153(X)) -> U153#(active(X))
   active#(U161(X1,X2)) -> active#(X1)
   active#(U161(X1,X2)) -> U161#(active(X1),X2)
   active#(cons(X1,X2)) -> active#(X1)
   active#(cons(X1,X2)) -> cons#(active(X1),X2)
   active#(natsFrom(X)) -> active#(X)
   active#(natsFrom(X)) -> natsFrom#(active(X))
   active#(s(X)) -> active#(X)
   active#(s(X)) -> s#(active(X))
   active#(U171(X1,X2,X3)) -> active#(X1)
   active#(U171(X1,X2,X3)) -> U171#(active(X1),X2,X3)
   active#(head(X)) -> active#(X)
   active#(head(X)) -> head#(active(X))
   active#(afterNth(X1,X2)) -> active#(X1)
   active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2)
   active#(afterNth(X1,X2)) -> active#(X2)
   active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2))
   active#(U181(X1,X2)) -> active#(X1)
   active#(U181(X1,X2)) -> U181#(active(X1),X2)
   active#(U191(X1,X2)) -> active#(X1)
   active#(U191(X1,X2)) -> U191#(active(X1),X2)
   active#(pair(X1,X2)) -> active#(X1)
   active#(pair(X1,X2)) -> pair#(active(X1),X2)
   active#(pair(X1,X2)) -> active#(X2)
   active#(pair(X1,X2)) -> pair#(X1,active(X2))
   active#(U201(X1,X2,X3,X4)) -> active#(X1)
   active#(U201(X1,X2,X3,X4)) -> U201#(active(X1),X2,X3,X4)
   active#(U202(X1,X2)) -> active#(X1)
   active#(U202(X1,X2)) -> U202#(active(X1),X2)
   active#(U21(X1,X2)) -> active#(X1)
   active#(U21(X1,X2)) -> U21#(active(X1),X2)
   active#(U211(X1,X2)) -> active#(X1)
   active#(U211(X1,X2)) -> U211#(active(X1),X2)
   active#(U221(X1,X2,X3)) -> active#(X1)
   active#(U221(X1,X2,X3)) -> U221#(active(X1),X2,X3)
   active#(fst(X)) -> active#(X)
   active#(fst(X)) -> fst#(active(X))
   active#(U31(X1,X2)) -> active#(X1)
   active#(U31(X1,X2)) -> U31#(active(X1),X2)
   active#(U41(X1,X2,X3)) -> active#(X1)
   active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
   active#(U42(X1,X2)) -> active#(X1)
   active#(U42(X1,X2)) -> U42#(active(X1),X2)
   active#(U43(X)) -> active#(X)
   active#(U43(X)) -> U43#(active(X))
   active#(U51(X1,X2,X3)) -> active#(X1)
   active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3)
   active#(U52(X1,X2)) -> active#(X1)
   active#(U52(X1,X2)) -> U52#(active(X1),X2)
   active#(U53(X)) -> active#(X)
   active#(U53(X)) -> U53#(active(X))
   active#(U61(X1,X2)) -> active#(X1)
   active#(U61(X1,X2)) -> U61#(active(X1),X2)
   active#(U62(X)) -> active#(X)
   active#(U62(X)) -> U62#(active(X))
   active#(U71(X1,X2)) -> active#(X1)
   active#(U71(X1,X2)) -> U71#(active(X1),X2)
   active#(U72(X)) -> active#(X)
   active#(U72(X)) -> U72#(active(X))
   active#(U81(X1,X2)) -> active#(X1)
   active#(U81(X1,X2)) -> U81#(active(X1),X2)
   active#(U82(X)) -> active#(X)
   active#(U82(X)) -> U82#(active(X))
   active#(U91(X1,X2)) -> active#(X1)
   active#(U91(X1,X2)) -> U91#(active(X1),X2)
   active#(U92(X)) -> active#(X)
   active#(U92(X)) -> U92#(active(X))
   active#(and(X1,X2)) -> active#(X1)
   active#(and(X1,X2)) -> and#(active(X1),X2)
   active#(tail(X)) -> active#(X)
   active#(tail(X)) -> tail#(active(X))
   active#(take(X1,X2)) -> active#(X1)
   active#(take(X1,X2)) -> take#(active(X1),X2)
   active#(take(X1,X2)) -> active#(X2)
   active#(take(X1,X2)) -> take#(X1,active(X2))
   active#(sel(X1,X2)) -> active#(X1)
   active#(sel(X1,X2)) -> sel#(active(X1),X2)
   active#(sel(X1,X2)) -> active#(X2)
   active#(sel(X1,X2)) -> sel#(X1,active(X2))
   U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
   U102#(mark(X1),X2) -> U102#(X1,X2)
   U103#(mark(X)) -> U103#(X)
   U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
   snd#(mark(X)) -> snd#(X)
   splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
   splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
   U111#(mark(X1),X2) -> U111#(X1,X2)
   U112#(mark(X)) -> U112#(X)
   U121#(mark(X1),X2) -> U121#(X1,X2)
   U122#(mark(X)) -> U122#(X)
   U131#(mark(X1),X2,X3) -> U131#(X1,X2,X3)
   U132#(mark(X1),X2) -> U132#(X1,X2)
   U133#(mark(X)) -> U133#(X)
   U141#(mark(X1),X2,X3) -> U141#(X1,X2,X3)
   U142#(mark(X1),X2) -> U142#(X1,X2)
   U143#(mark(X)) -> U143#(X)
   U151#(mark(X1),X2,X3) -> U151#(X1,X2,X3)
   U152#(mark(X1),X2) -> U152#(X1,X2)
   U153#(mark(X)) -> U153#(X)
   U161#(mark(X1),X2) -> U161#(X1,X2)
   cons#(mark(X1),X2) -> cons#(X1,X2)
   natsFrom#(mark(X)) -> natsFrom#(X)
   s#(mark(X)) -> s#(X)
   U171#(mark(X1),X2,X3) -> U171#(X1,X2,X3)
   head#(mark(X)) -> head#(X)
   afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
   afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
   U181#(mark(X1),X2) -> U181#(X1,X2)
   U191#(mark(X1),X2) -> U191#(X1,X2)
   pair#(mark(X1),X2) -> pair#(X1,X2)
   pair#(X1,mark(X2)) -> pair#(X1,X2)
   U201#(mark(X1),X2,X3,X4) -> U201#(X1,X2,X3,X4)
   U202#(mark(X1),X2) -> U202#(X1,X2)
   U21#(mark(X1),X2) -> U21#(X1,X2)
   U211#(mark(X1),X2) -> U211#(X1,X2)
   U221#(mark(X1),X2,X3) -> U221#(X1,X2,X3)
   fst#(mark(X)) -> fst#(X)
   U31#(mark(X1),X2) -> U31#(X1,X2)
   U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
   U42#(mark(X1),X2) -> U42#(X1,X2)
   U43#(mark(X)) -> U43#(X)
   U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
   U52#(mark(X1),X2) -> U52#(X1,X2)
   U53#(mark(X)) -> U53#(X)
   U61#(mark(X1),X2) -> U61#(X1,X2)
   U62#(mark(X)) -> U62#(X)
   U71#(mark(X1),X2) -> U71#(X1,X2)
   U72#(mark(X)) -> U72#(X)
   U81#(mark(X1),X2) -> U81#(X1,X2)
   U82#(mark(X)) -> U82#(X)
   U91#(mark(X1),X2) -> U91#(X1,X2)
   U92#(mark(X)) -> U92#(X)
   and#(mark(X1),X2) -> and#(X1,X2)
   tail#(mark(X)) -> tail#(X)
   take#(mark(X1),X2) -> take#(X1,X2)
   take#(X1,mark(X2)) -> take#(X1,X2)
   sel#(mark(X1),X2) -> sel#(X1,X2)
   sel#(X1,mark(X2)) -> sel#(X1,X2)
   proper#(U101(X1,X2,X3)) -> proper#(X3)
   proper#(U101(X1,X2,X3)) -> proper#(X2)
   proper#(U101(X1,X2,X3)) -> proper#(X1)
   proper#(U101(X1,X2,X3)) -> U101#(proper(X1),proper(X2),proper(X3))
   proper#(U102(X1,X2)) -> proper#(X2)
   proper#(U102(X1,X2)) -> proper#(X1)
   proper#(U102(X1,X2)) -> U102#(proper(X1),proper(X2))
   proper#(isNatural(X)) -> proper#(X)
   proper#(isNatural(X)) -> isNatural#(proper(X))
   proper#(U103(X)) -> proper#(X)
   proper#(U103(X)) -> U103#(proper(X))
   proper#(isLNat(X)) -> proper#(X)
   proper#(isLNat(X)) -> isLNat#(proper(X))
   proper#(U11(X1,X2,X3)) -> proper#(X3)
   proper#(U11(X1,X2,X3)) -> proper#(X2)
   proper#(U11(X1,X2,X3)) -> proper#(X1)
   proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
   proper#(snd(X)) -> proper#(X)
   proper#(snd(X)) -> snd#(proper(X))
   proper#(splitAt(X1,X2)) -> proper#(X2)
   proper#(splitAt(X1,X2)) -> proper#(X1)
   proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2))
   proper#(U111(X1,X2)) -> proper#(X2)
   proper#(U111(X1,X2)) -> proper#(X1)
   proper#(U111(X1,X2)) -> U111#(proper(X1),proper(X2))
   proper#(U112(X)) -> proper#(X)
   proper#(U112(X)) -> U112#(proper(X))
   proper#(U121(X1,X2)) -> proper#(X2)
   proper#(U121(X1,X2)) -> proper#(X1)
   proper#(U121(X1,X2)) -> U121#(proper(X1),proper(X2))
   proper#(U122(X)) -> proper#(X)
   proper#(U122(X)) -> U122#(proper(X))
   proper#(U131(X1,X2,X3)) -> proper#(X3)
   proper#(U131(X1,X2,X3)) -> proper#(X2)
   proper#(U131(X1,X2,X3)) -> proper#(X1)
   proper#(U131(X1,X2,X3)) -> U131#(proper(X1),proper(X2),proper(X3))
   proper#(U132(X1,X2)) -> proper#(X2)
   proper#(U132(X1,X2)) -> proper#(X1)
   proper#(U132(X1,X2)) -> U132#(proper(X1),proper(X2))
   proper#(U133(X)) -> proper#(X)
   proper#(U133(X)) -> U133#(proper(X))
   proper#(U141(X1,X2,X3)) -> proper#(X3)
   proper#(U141(X1,X2,X3)) -> proper#(X2)
   proper#(U141(X1,X2,X3)) -> proper#(X1)
   proper#(U141(X1,X2,X3)) -> U141#(proper(X1),proper(X2),proper(X3))
   proper#(U142(X1,X2)) -> proper#(X2)
   proper#(U142(X1,X2)) -> proper#(X1)
   proper#(U142(X1,X2)) -> U142#(proper(X1),proper(X2))
   proper#(U143(X)) -> proper#(X)
   proper#(U143(X)) -> U143#(proper(X))
   proper#(U151(X1,X2,X3)) -> proper#(X3)
   proper#(U151(X1,X2,X3)) -> proper#(X2)
   proper#(U151(X1,X2,X3)) -> proper#(X1)
   proper#(U151(X1,X2,X3)) -> U151#(proper(X1),proper(X2),proper(X3))
   proper#(U152(X1,X2)) -> proper#(X2)
   proper#(U152(X1,X2)) -> proper#(X1)
   proper#(U152(X1,X2)) -> U152#(proper(X1),proper(X2))
   proper#(U153(X)) -> proper#(X)
   proper#(U153(X)) -> U153#(proper(X))
   proper#(U161(X1,X2)) -> proper#(X2)
   proper#(U161(X1,X2)) -> proper#(X1)
   proper#(U161(X1,X2)) -> U161#(proper(X1),proper(X2))
   proper#(cons(X1,X2)) -> proper#(X2)
   proper#(cons(X1,X2)) -> proper#(X1)
   proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
   proper#(natsFrom(X)) -> proper#(X)
   proper#(natsFrom(X)) -> natsFrom#(proper(X))
   proper#(s(X)) -> proper#(X)
   proper#(s(X)) -> s#(proper(X))
   proper#(U171(X1,X2,X3)) -> proper#(X3)
   proper#(U171(X1,X2,X3)) -> proper#(X2)
   proper#(U171(X1,X2,X3)) -> proper#(X1)
   proper#(U171(X1,X2,X3)) -> U171#(proper(X1),proper(X2),proper(X3))
   proper#(head(X)) -> proper#(X)
   proper#(head(X)) -> head#(proper(X))
   proper#(afterNth(X1,X2)) -> proper#(X2)
   proper#(afterNth(X1,X2)) -> proper#(X1)
   proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2))
   proper#(U181(X1,X2)) -> proper#(X2)
   proper#(U181(X1,X2)) -> proper#(X1)
   proper#(U181(X1,X2)) -> U181#(proper(X1),proper(X2))
   proper#(U191(X1,X2)) -> proper#(X2)
   proper#(U191(X1,X2)) -> proper#(X1)
   proper#(U191(X1,X2)) -> U191#(proper(X1),proper(X2))
   proper#(pair(X1,X2)) -> proper#(X2)
   proper#(pair(X1,X2)) -> proper#(X1)
   proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2))
   proper#(U201(X1,X2,X3,X4)) -> proper#(X4)
   proper#(U201(X1,X2,X3,X4)) -> proper#(X3)
   proper#(U201(X1,X2,X3,X4)) -> proper#(X2)
   proper#(U201(X1,X2,X3,X4)) -> proper#(X1)
   proper#(U201(X1,X2,X3,X4)) -> U201#(proper(X1),proper(X2),proper(X3),proper(X4))
   proper#(U202(X1,X2)) -> proper#(X2)
   proper#(U202(X1,X2)) -> proper#(X1)
   proper#(U202(X1,X2)) -> U202#(proper(X1),proper(X2))
   proper#(U21(X1,X2)) -> proper#(X2)
   proper#(U21(X1,X2)) -> proper#(X1)
   proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
   proper#(U211(X1,X2)) -> proper#(X2)
   proper#(U211(X1,X2)) -> proper#(X1)
   proper#(U211(X1,X2)) -> U211#(proper(X1),proper(X2))
   proper#(U221(X1,X2,X3)) -> proper#(X3)
   proper#(U221(X1,X2,X3)) -> proper#(X2)
   proper#(U221(X1,X2,X3)) -> proper#(X1)
   proper#(U221(X1,X2,X3)) -> U221#(proper(X1),proper(X2),proper(X3))
   proper#(fst(X)) -> proper#(X)
   proper#(fst(X)) -> fst#(proper(X))
   proper#(U31(X1,X2)) -> proper#(X2)
   proper#(U31(X1,X2)) -> proper#(X1)
   proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
   proper#(U41(X1,X2,X3)) -> proper#(X3)
   proper#(U41(X1,X2,X3)) -> proper#(X2)
   proper#(U41(X1,X2,X3)) -> proper#(X1)
   proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
   proper#(U42(X1,X2)) -> proper#(X2)
   proper#(U42(X1,X2)) -> proper#(X1)
   proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2))
   proper#(U43(X)) -> proper#(X)
   proper#(U43(X)) -> U43#(proper(X))
   proper#(U51(X1,X2,X3)) -> proper#(X3)
   proper#(U51(X1,X2,X3)) -> proper#(X2)
   proper#(U51(X1,X2,X3)) -> proper#(X1)
   proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3))
   proper#(U52(X1,X2)) -> proper#(X2)
   proper#(U52(X1,X2)) -> proper#(X1)
   proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2))
   proper#(U53(X)) -> proper#(X)
   proper#(U53(X)) -> U53#(proper(X))
   proper#(U61(X1,X2)) -> proper#(X2)
   proper#(U61(X1,X2)) -> proper#(X1)
   proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2))
   proper#(U62(X)) -> proper#(X)
   proper#(U62(X)) -> U62#(proper(X))
   proper#(isPLNat(X)) -> proper#(X)
   proper#(isPLNat(X)) -> isPLNat#(proper(X))
   proper#(U71(X1,X2)) -> proper#(X2)
   proper#(U71(X1,X2)) -> proper#(X1)
   proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2))
   proper#(U72(X)) -> proper#(X)
   proper#(U72(X)) -> U72#(proper(X))
   proper#(U81(X1,X2)) -> proper#(X2)
   proper#(U81(X1,X2)) -> proper#(X1)
   proper#(U81(X1,X2)) -> U81#(proper(X1),proper(X2))
   proper#(U82(X)) -> proper#(X)
   proper#(U82(X)) -> U82#(proper(X))
   proper#(U91(X1,X2)) -> proper#(X2)
   proper#(U91(X1,X2)) -> proper#(X1)
   proper#(U91(X1,X2)) -> U91#(proper(X1),proper(X2))
   proper#(U92(X)) -> proper#(X)
   proper#(U92(X)) -> U92#(proper(X))
   proper#(and(X1,X2)) -> proper#(X2)
   proper#(and(X1,X2)) -> proper#(X1)
   proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
   proper#(isNaturalKind(X)) -> proper#(X)
   proper#(isNaturalKind(X)) -> isNaturalKind#(proper(X))
   proper#(isLNatKind(X)) -> proper#(X)
   proper#(isLNatKind(X)) -> isLNatKind#(proper(X))
   proper#(isPLNatKind(X)) -> proper#(X)
   proper#(isPLNatKind(X)) -> isPLNatKind#(proper(X))
   proper#(tail(X)) -> proper#(X)
   proper#(tail(X)) -> tail#(proper(X))
   proper#(take(X1,X2)) -> proper#(X2)
   proper#(take(X1,X2)) -> proper#(X1)
   proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
   proper#(sel(X1,X2)) -> proper#(X2)
   proper#(sel(X1,X2)) -> proper#(X1)
   proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2))
   U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3)
   U102#(ok(X1),ok(X2)) -> U102#(X1,X2)
   isNatural#(ok(X)) -> isNatural#(X)
   U103#(ok(X)) -> U103#(X)
   isLNat#(ok(X)) -> isLNat#(X)
   U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
   snd#(ok(X)) -> snd#(X)
   splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
   U111#(ok(X1),ok(X2)) -> U111#(X1,X2)
   U112#(ok(X)) -> U112#(X)
   U121#(ok(X1),ok(X2)) -> U121#(X1,X2)
   U122#(ok(X)) -> U122#(X)
   U131#(ok(X1),ok(X2),ok(X3)) -> U131#(X1,X2,X3)
   U132#(ok(X1),ok(X2)) -> U132#(X1,X2)
   U133#(ok(X)) -> U133#(X)
   U141#(ok(X1),ok(X2),ok(X3)) -> U141#(X1,X2,X3)
   U142#(ok(X1),ok(X2)) -> U142#(X1,X2)
   U143#(ok(X)) -> U143#(X)
   U151#(ok(X1),ok(X2),ok(X3)) -> U151#(X1,X2,X3)
   U152#(ok(X1),ok(X2)) -> U152#(X1,X2)
   U153#(ok(X)) -> U153#(X)
   U161#(ok(X1),ok(X2)) -> U161#(X1,X2)
   cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
   natsFrom#(ok(X)) -> natsFrom#(X)
   s#(ok(X)) -> s#(X)
   U171#(ok(X1),ok(X2),ok(X3)) -> U171#(X1,X2,X3)
   head#(ok(X)) -> head#(X)
   afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
   U181#(ok(X1),ok(X2)) -> U181#(X1,X2)
   U191#(ok(X1),ok(X2)) -> U191#(X1,X2)
   pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
   U201#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U201#(X1,X2,X3,X4)
   U202#(ok(X1),ok(X2)) -> U202#(X1,X2)
   U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
   U211#(ok(X1),ok(X2)) -> U211#(X1,X2)
   U221#(ok(X1),ok(X2),ok(X3)) -> U221#(X1,X2,X3)
   fst#(ok(X)) -> fst#(X)
   U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
   U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
   U42#(ok(X1),ok(X2)) -> U42#(X1,X2)
   U43#(ok(X)) -> U43#(X)
   U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
   U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
   U53#(ok(X)) -> U53#(X)
   U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
   U62#(ok(X)) -> U62#(X)
   isPLNat#(ok(X)) -> isPLNat#(X)
   U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
   U72#(ok(X)) -> U72#(X)
   U81#(ok(X1),ok(X2)) -> U81#(X1,X2)
   U82#(ok(X)) -> U82#(X)
   U91#(ok(X1),ok(X2)) -> U91#(X1,X2)
   U92#(ok(X)) -> U92#(X)
   and#(ok(X1),ok(X2)) -> and#(X1,X2)
   isNaturalKind#(ok(X)) -> isNaturalKind#(X)
   isLNatKind#(ok(X)) -> isLNatKind#(X)
   isPLNatKind#(ok(X)) -> isPLNatKind#(X)
   tail#(ok(X)) -> tail#(X)
   take#(ok(X1),ok(X2)) -> take#(X1,X2)
   sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
   active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
   active(U103(tt())) -> mark(tt())
   active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
   active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
   active(U112(tt())) -> mark(tt())
   active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
   active(U122(tt())) -> mark(tt())
   active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
   active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
   active(U133(tt())) -> mark(tt())
   active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
   active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
   active(U143(tt())) -> mark(tt())
   active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
   active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
   active(U153(tt())) -> mark(tt())
   active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
   active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
   active(U181(tt(),Y)) -> mark(Y)
   active(U191(tt(),XS)) -> mark(pair(nil(),XS))
   active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
   active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
   active(U21(tt(),X)) -> mark(X)
   active(U211(tt(),XS)) -> mark(XS)
   active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
   active(U31(tt(),N)) -> mark(N)
   active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
   active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
   active(U43(tt())) -> mark(tt())
   active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
   active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
   active(U53(tt())) -> mark(tt())
   active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
   active(U62(tt())) -> mark(tt())
   active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
   active(U72(tt())) -> mark(tt())
   active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
   active(U82(tt())) -> mark(tt())
   active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
   active(U92(tt())) -> mark(tt())
   active(afterNth(N,XS)) ->
   mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
   active(and(tt(),X)) -> mark(X)
   active(fst(pair(X,Y))) ->
   mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
   active(head(cons(N,XS))) ->
   mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
   active(isLNat(nil())) -> mark(tt())
   active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
   active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
   active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
   active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
   active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
   active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
   active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
   active(isLNatKind(nil())) -> mark(tt())
   active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
   active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
   active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
   active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
   active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
   active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
   active(isNatural(0())) -> mark(tt())
   active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
   active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
   active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
   active(isNaturalKind(0())) -> mark(tt())
   active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
   active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
   active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
   active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
   active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
   active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
   active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
   active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
   active(sel(N,XS)) ->
   mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
   active(snd(pair(X,Y))) ->
   mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
   active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
   active(splitAt(s(N),cons(X,XS))) ->
   mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                        and(isLNat(XS),isLNatKind(XS)))),
             N,X,XS))
   active(tail(cons(N,XS))) ->
   mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
   active(take(N,XS)) ->
   mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
   active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
   active(U102(X1,X2)) -> U102(active(X1),X2)
   active(U103(X)) -> U103(active(X))
   active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
   active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
   active(U111(X1,X2)) -> U111(active(X1),X2)
   active(U112(X)) -> U112(active(X))
   active(U121(X1,X2)) -> U121(active(X1),X2)
   active(U122(X)) -> U122(active(X))
   active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
   active(U132(X1,X2)) -> U132(active(X1),X2)
   active(U133(X)) -> U133(active(X))
   active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
   active(U142(X1,X2)) -> U142(active(X1),X2)
   active(U143(X)) -> U143(active(X))
   active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
   active(U152(X1,X2)) -> U152(active(X1),X2)
   active(U153(X)) -> U153(active(X))
   active(U161(X1,X2)) -> U161(active(X1),X2)
   active(cons(X1,X2)) -> cons(active(X1),X2)
   active(natsFrom(X)) -> natsFrom(active(X))
   active(s(X)) -> s(active(X))
   active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
   active(head(X)) -> head(active(X))
   active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
   active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
   active(U181(X1,X2)) -> U181(active(X1),X2)
   active(U191(X1,X2)) -> U191(active(X1),X2)
   active(pair(X1,X2)) -> pair(active(X1),X2)
   active(pair(X1,X2)) -> pair(X1,active(X2))
   active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
   active(U202(X1,X2)) -> U202(active(X1),X2)
   active(U21(X1,X2)) -> U21(active(X1),X2)
   active(U211(X1,X2)) -> U211(active(X1),X2)
   active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
   active(fst(X)) -> fst(active(X))
   active(U31(X1,X2)) -> U31(active(X1),X2)
   active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
   active(U42(X1,X2)) -> U42(active(X1),X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
   active(U52(X1,X2)) -> U52(active(X1),X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1,X2)) -> U61(active(X1),X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1,X2)) -> U71(active(X1),X2)
   active(U72(X)) -> U72(active(X))
   active(U81(X1,X2)) -> U81(active(X1),X2)
   active(U82(X)) -> U82(active(X))
   active(U91(X1,X2)) -> U91(active(X1),X2)
   active(U92(X)) -> U92(active(X))
   active(and(X1,X2)) -> and(active(X1),X2)
   active(tail(X)) -> tail(active(X))
   active(take(X1,X2)) -> take(active(X1),X2)
   active(take(X1,X2)) -> take(X1,active(X2))
   active(sel(X1,X2)) -> sel(active(X1),X2)
   active(sel(X1,X2)) -> sel(X1,active(X2))
   U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
   U102(mark(X1),X2) -> mark(U102(X1,X2))
   U103(mark(X)) -> mark(U103(X))
   U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
   splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
   U111(mark(X1),X2) -> mark(U111(X1,X2))
   U112(mark(X)) -> mark(U112(X))
   U121(mark(X1),X2) -> mark(U121(X1,X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
   U132(mark(X1),X2) -> mark(U132(X1,X2))
   U133(mark(X)) -> mark(U133(X))
   U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
   U142(mark(X1),X2) -> mark(U142(X1,X2))
   U143(mark(X)) -> mark(U143(X))
   U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
   U152(mark(X1),X2) -> mark(U152(X1,X2))
   U153(mark(X)) -> mark(U153(X))
   U161(mark(X1),X2) -> mark(U161(X1,X2))
   cons(mark(X1),X2) -> mark(cons(X1,X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   s(mark(X)) -> mark(s(X))
   U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
   head(mark(X)) -> mark(head(X))
   afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
   afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
   U181(mark(X1),X2) -> mark(U181(X1,X2))
   U191(mark(X1),X2) -> mark(U191(X1,X2))
   pair(mark(X1),X2) -> mark(pair(X1,X2))
   pair(X1,mark(X2)) -> mark(pair(X1,X2))
   U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
   U202(mark(X1),X2) -> mark(U202(X1,X2))
   U21(mark(X1),X2) -> mark(U21(X1,X2))
   U211(mark(X1),X2) -> mark(U211(X1,X2))
   U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
   fst(mark(X)) -> mark(fst(X))
   U31(mark(X1),X2) -> mark(U31(X1,X2))
   U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
   U42(mark(X1),X2) -> mark(U42(X1,X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
   U52(mark(X1),X2) -> mark(U52(X1,X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1),X2) -> mark(U61(X1,X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1),X2) -> mark(U71(X1,X2))
   U72(mark(X)) -> mark(U72(X))
   U81(mark(X1),X2) -> mark(U81(X1,X2))
   U82(mark(X)) -> mark(U82(X))
   U91(mark(X1),X2) -> mark(U91(X1,X2))
   U92(mark(X)) -> mark(U92(X))
   and(mark(X1),X2) -> mark(and(X1,X2))
   tail(mark(X)) -> mark(tail(X))
   take(mark(X1),X2) -> mark(take(X1,X2))
   take(X1,mark(X2)) -> mark(take(X1,X2))
   sel(mark(X1),X2) -> mark(sel(X1,X2))
   sel(X1,mark(X2)) -> mark(sel(X1,X2))
   proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
   proper(tt()) -> ok(tt())
   proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
   proper(isNatural(X)) -> isNatural(proper(X))
   proper(U103(X)) -> U103(proper(X))
   proper(isLNat(X)) -> isLNat(proper(X))
   proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
   proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
   proper(U112(X)) -> U112(proper(X))
   proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
   proper(U122(X)) -> U122(proper(X))
   proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
   proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
   proper(U133(X)) -> U133(proper(X))
   proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
   proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
   proper(U143(X)) -> U143(proper(X))
   proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
   proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
   proper(U153(X)) -> U153(proper(X))
   proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
   proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
   proper(head(X)) -> head(proper(X))
   proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
   proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
   proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
   proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
   proper(nil()) -> ok(nil())
   proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
   proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
   proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
   proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
   proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
   proper(fst(X)) -> fst(proper(X))
   proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
   proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
   proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
   proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(isPLNat(X)) -> isPLNat(proper(X))
   proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
   proper(U82(X)) -> U82(proper(X))
   proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
   proper(U92(X)) -> U92(proper(X))
   proper(and(X1,X2)) -> and(proper(X1),proper(X2))
   proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
   proper(isLNatKind(X)) -> isLNatKind(proper(X))
   proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(take(X1,X2)) -> take(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
   U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
   U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
   isNatural(ok(X)) -> ok(isNatural(X))
   U103(ok(X)) -> ok(U103(X))
   isLNat(ok(X)) -> ok(isLNat(X))
   U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
   U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
   U112(ok(X)) -> ok(U112(X))
   U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
   U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
   U133(ok(X)) -> ok(U133(X))
   U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
   U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
   U143(ok(X)) -> ok(U143(X))
   U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
   U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
   U153(ok(X)) -> ok(U153(X))
   U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
   cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   s(ok(X)) -> ok(s(X))
   U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
   head(ok(X)) -> ok(head(X))
   afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
   U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
   U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
   pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
   U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
   U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
   U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
   U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
   U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
   fst(ok(X)) -> ok(fst(X))
   U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
   U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
   U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
   U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
   U62(ok(X)) -> ok(U62(X))
   isPLNat(ok(X)) -> ok(isPLNat(X))
   U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
   U72(ok(X)) -> ok(U72(X))
   U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
   U82(ok(X)) -> ok(U82(X))
   U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
   U92(ok(X)) -> ok(U92(X))
   and(ok(X1),ok(X2)) -> ok(and(X1,X2))
   isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
   isLNatKind(ok(X)) -> ok(isLNatKind(X))
   isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
   tail(ok(X)) -> ok(tail(X))
   take(ok(X1),ok(X2)) -> ok(take(X1,X2))
   sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  Usable Rule Processor:
   DPs:
    active#(U101(tt(),V1,V2)) -> isNatural#(V1)
    active#(U101(tt(),V1,V2)) -> U102#(isNatural(V1),V2)
    active#(U102(tt(),V2)) -> isLNat#(V2)
    active#(U102(tt(),V2)) -> U103#(isLNat(V2))
    active#(U11(tt(),N,XS)) -> splitAt#(N,XS)
    active#(U11(tt(),N,XS)) -> snd#(splitAt(N,XS))
    active#(U111(tt(),V1)) -> isLNat#(V1)
    active#(U111(tt(),V1)) -> U112#(isLNat(V1))
    active#(U121(tt(),V1)) -> isNatural#(V1)
    active#(U121(tt(),V1)) -> U122#(isNatural(V1))
    active#(U131(tt(),V1,V2)) -> isNatural#(V1)
    active#(U131(tt(),V1,V2)) -> U132#(isNatural(V1),V2)
    active#(U132(tt(),V2)) -> isLNat#(V2)
    active#(U132(tt(),V2)) -> U133#(isLNat(V2))
    active#(U141(tt(),V1,V2)) -> isLNat#(V1)
    active#(U141(tt(),V1,V2)) -> U142#(isLNat(V1),V2)
    active#(U142(tt(),V2)) -> isLNat#(V2)
    active#(U142(tt(),V2)) -> U143#(isLNat(V2))
    active#(U151(tt(),V1,V2)) -> isNatural#(V1)
    active#(U151(tt(),V1,V2)) -> U152#(isNatural(V1),V2)
    active#(U152(tt(),V2)) -> isLNat#(V2)
    active#(U152(tt(),V2)) -> U153#(isLNat(V2))
    active#(U161(tt(),N)) -> s#(N)
    active#(U161(tt(),N)) -> natsFrom#(s(N))
    active#(U161(tt(),N)) -> cons#(N,natsFrom(s(N)))
    active#(U171(tt(),N,XS)) -> afterNth#(N,XS)
    active#(U171(tt(),N,XS)) -> head#(afterNth(N,XS))
    active#(U191(tt(),XS)) -> pair#(nil(),XS)
    active#(U201(tt(),N,X,XS)) -> splitAt#(N,XS)
    active#(U201(tt(),N,X,XS)) -> U202#(splitAt(N,XS),X)
    active#(U202(pair(YS,ZS),X)) -> cons#(X,YS)
    active#(U202(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS)
    active#(U221(tt(),N,XS)) -> splitAt#(N,XS)
    active#(U221(tt(),N,XS)) -> fst#(splitAt(N,XS))
    active#(U41(tt(),V1,V2)) -> isNatural#(V1)
    active#(U41(tt(),V1,V2)) -> U42#(isNatural(V1),V2)
    active#(U42(tt(),V2)) -> isLNat#(V2)
    active#(U42(tt(),V2)) -> U43#(isLNat(V2))
    active#(U51(tt(),V1,V2)) -> isNatural#(V1)
    active#(U51(tt(),V1,V2)) -> U52#(isNatural(V1),V2)
    active#(U52(tt(),V2)) -> isLNat#(V2)
    active#(U52(tt(),V2)) -> U53#(isLNat(V2))
    active#(U61(tt(),V1)) -> isPLNat#(V1)
    active#(U61(tt(),V1)) -> U62#(isPLNat(V1))
    active#(U71(tt(),V1)) -> isNatural#(V1)
    active#(U71(tt(),V1)) -> U72#(isNatural(V1))
    active#(U81(tt(),V1)) -> isPLNat#(V1)
    active#(U81(tt(),V1)) -> U82#(isPLNat(V1))
    active#(U91(tt(),V1)) -> isLNat#(V1)
    active#(U91(tt(),V1)) -> U92#(isLNat(V1))
    active#(afterNth(N,XS)) -> isLNatKind#(XS)
    active#(afterNth(N,XS)) -> isLNat#(XS)
    active#(afterNth(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
    active#(afterNth(N,XS)) -> isNaturalKind#(N)
    active#(afterNth(N,XS)) -> isNatural#(N)
    active#(afterNth(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
    active#(afterNth(N,XS)) ->
    and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
    active#(afterNth(N,XS)) ->
    U11#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
    active#(fst(pair(X,Y))) -> isLNatKind#(Y)
    active#(fst(pair(X,Y))) -> isLNat#(Y)
    active#(fst(pair(X,Y))) -> and#(isLNat(Y),isLNatKind(Y))
    active#(fst(pair(X,Y))) -> isLNatKind#(X)
    active#(fst(pair(X,Y))) -> isLNat#(X)
    active#(fst(pair(X,Y))) -> and#(isLNat(X),isLNatKind(X))
    active#(fst(pair(X,Y))) -> and#(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y)))
    active#(fst(pair(X,Y))) ->
    U21#(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X)
    active#(head(cons(N,XS))) -> isLNatKind#(XS)
    active#(head(cons(N,XS))) -> isLNat#(XS)
    active#(head(cons(N,XS))) -> and#(isLNat(XS),isLNatKind(XS))
    active#(head(cons(N,XS))) -> isNaturalKind#(N)
    active#(head(cons(N,XS))) -> isNatural#(N)
    active#(head(cons(N,XS))) -> and#(isNatural(N),isNaturalKind(N))
    active#(head(cons(N,XS))) ->
    and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
    active#(head(cons(N,XS))) ->
    U31#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N)
    active#(isLNat(afterNth(V1,V2))) -> isLNatKind#(V2)
    active#(isLNat(afterNth(V1,V2))) -> isNaturalKind#(V1)
    active#(isLNat(afterNth(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isLNat(afterNth(V1,V2))) -> U41#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
    active#(isLNat(cons(V1,V2))) -> isLNatKind#(V2)
    active#(isLNat(cons(V1,V2))) -> isNaturalKind#(V1)
    active#(isLNat(cons(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isLNat(cons(V1,V2))) -> U51#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
    active#(isLNat(fst(V1))) -> isPLNatKind#(V1)
    active#(isLNat(fst(V1))) -> U61#(isPLNatKind(V1),V1)
    active#(isLNat(natsFrom(V1))) -> isNaturalKind#(V1)
    active#(isLNat(natsFrom(V1))) -> U71#(isNaturalKind(V1),V1)
    active#(isLNat(snd(V1))) -> isPLNatKind#(V1)
    active#(isLNat(snd(V1))) -> U81#(isPLNatKind(V1),V1)
    active#(isLNat(tail(V1))) -> isLNatKind#(V1)
    active#(isLNat(tail(V1))) -> U91#(isLNatKind(V1),V1)
    active#(isLNat(take(V1,V2))) -> isLNatKind#(V2)
    active#(isLNat(take(V1,V2))) -> isNaturalKind#(V1)
    active#(isLNat(take(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isLNat(take(V1,V2))) -> U101#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
    active#(isLNatKind(afterNth(V1,V2))) -> isLNatKind#(V2)
    active#(isLNatKind(afterNth(V1,V2))) -> isNaturalKind#(V1)
    active#(isLNatKind(afterNth(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isLNatKind(cons(V1,V2))) -> isLNatKind#(V2)
    active#(isLNatKind(cons(V1,V2))) -> isNaturalKind#(V1)
    active#(isLNatKind(cons(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isLNatKind(fst(V1))) -> isPLNatKind#(V1)
    active#(isLNatKind(natsFrom(V1))) -> isNaturalKind#(V1)
    active#(isLNatKind(snd(V1))) -> isPLNatKind#(V1)
    active#(isLNatKind(tail(V1))) -> isLNatKind#(V1)
    active#(isLNatKind(take(V1,V2))) -> isLNatKind#(V2)
    active#(isLNatKind(take(V1,V2))) -> isNaturalKind#(V1)
    active#(isLNatKind(take(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isNatural(head(V1))) -> isLNatKind#(V1)
    active#(isNatural(head(V1))) -> U111#(isLNatKind(V1),V1)
    active#(isNatural(s(V1))) -> isNaturalKind#(V1)
    active#(isNatural(s(V1))) -> U121#(isNaturalKind(V1),V1)
    active#(isNatural(sel(V1,V2))) -> isLNatKind#(V2)
    active#(isNatural(sel(V1,V2))) -> isNaturalKind#(V1)
    active#(isNatural(sel(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isNatural(sel(V1,V2))) -> U131#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
    active#(isNaturalKind(head(V1))) -> isLNatKind#(V1)
    active#(isNaturalKind(s(V1))) -> isNaturalKind#(V1)
    active#(isNaturalKind(sel(V1,V2))) -> isLNatKind#(V2)
    active#(isNaturalKind(sel(V1,V2))) -> isNaturalKind#(V1)
    active#(isNaturalKind(sel(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isPLNat(pair(V1,V2))) -> isLNatKind#(V2)
    active#(isPLNat(pair(V1,V2))) -> isLNatKind#(V1)
    active#(isPLNat(pair(V1,V2))) -> and#(isLNatKind(V1),isLNatKind(V2))
    active#(isPLNat(pair(V1,V2))) -> U141#(and(isLNatKind(V1),isLNatKind(V2)),V1,V2)
    active#(isPLNat(splitAt(V1,V2))) -> isLNatKind#(V2)
    active#(isPLNat(splitAt(V1,V2))) -> isNaturalKind#(V1)
    active#(isPLNat(splitAt(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(isPLNat(splitAt(V1,V2))) -> U151#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
    active#(isPLNatKind(pair(V1,V2))) -> isLNatKind#(V2)
    active#(isPLNatKind(pair(V1,V2))) -> isLNatKind#(V1)
    active#(isPLNatKind(pair(V1,V2))) -> and#(isLNatKind(V1),isLNatKind(V2))
    active#(isPLNatKind(splitAt(V1,V2))) -> isLNatKind#(V2)
    active#(isPLNatKind(splitAt(V1,V2))) -> isNaturalKind#(V1)
    active#(isPLNatKind(splitAt(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
    active#(natsFrom(N)) -> isNaturalKind#(N)
    active#(natsFrom(N)) -> isNatural#(N)
    active#(natsFrom(N)) -> and#(isNatural(N),isNaturalKind(N))
    active#(natsFrom(N)) -> U161#(and(isNatural(N),isNaturalKind(N)),N)
    active#(sel(N,XS)) -> isLNatKind#(XS)
    active#(sel(N,XS)) -> isLNat#(XS)
    active#(sel(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
    active#(sel(N,XS)) -> isNaturalKind#(N)
    active#(sel(N,XS)) -> isNatural#(N)
    active#(sel(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
    active#(sel(N,XS)) -> and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
    active#(sel(N,XS)) ->
    U171#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
    active#(snd(pair(X,Y))) -> isLNatKind#(Y)
    active#(snd(pair(X,Y))) -> isLNat#(Y)
    active#(snd(pair(X,Y))) -> and#(isLNat(Y),isLNatKind(Y))
    active#(snd(pair(X,Y))) -> isLNatKind#(X)
    active#(snd(pair(X,Y))) -> isLNat#(X)
    active#(snd(pair(X,Y))) -> and#(isLNat(X),isLNatKind(X))
    active#(snd(pair(X,Y))) -> and#(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y)))
    active#(snd(pair(X,Y))) ->
    U181#(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y)
    active#(splitAt(0(),XS)) -> isLNatKind#(XS)
    active#(splitAt(0(),XS)) -> isLNat#(XS)
    active#(splitAt(0(),XS)) -> and#(isLNat(XS),isLNatKind(XS))
    active#(splitAt(0(),XS)) -> U191#(and(isLNat(XS),isLNatKind(XS)),XS)
    active#(splitAt(s(N),cons(X,XS))) -> isLNatKind#(XS)
    active#(splitAt(s(N),cons(X,XS))) -> isLNat#(XS)
    active#(splitAt(s(N),cons(X,XS))) -> and#(isLNat(XS),isLNatKind(XS))
    active#(splitAt(s(N),cons(X,XS))) -> isNaturalKind#(X)
    active#(splitAt(s(N),cons(X,XS))) -> isNatural#(X)
    active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(X),isNaturalKind(X))
    active#(splitAt(s(N),cons(X,XS))) ->
    and#(and(isNatural(X),isNaturalKind(X)),and(isLNat(XS),isLNatKind(XS)))
    active#(splitAt(s(N),cons(X,XS))) -> isNaturalKind#(N)
    active#(splitAt(s(N),cons(X,XS))) -> isNatural#(N)
    active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(N),isNaturalKind(N))
    active#(splitAt(s(N),cons(X,XS))) ->
    and#(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                and(isLNat(XS),isLNatKind(XS))))
    active#(splitAt(s(N),cons(X,XS))) ->
    U201#(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                     and(isLNat(XS),isLNatKind(XS)))),
          N,X,XS)
    active#(tail(cons(N,XS))) -> isLNatKind#(XS)
    active#(tail(cons(N,XS))) -> isLNat#(XS)
    active#(tail(cons(N,XS))) -> and#(isLNat(XS),isLNatKind(XS))
    active#(tail(cons(N,XS))) -> isNaturalKind#(N)
    active#(tail(cons(N,XS))) -> isNatural#(N)
    active#(tail(cons(N,XS))) -> and#(isNatural(N),isNaturalKind(N))
    active#(tail(cons(N,XS))) ->
    and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
    active#(tail(cons(N,XS))) ->
    U211#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS)
    active#(take(N,XS)) -> isLNatKind#(XS)
    active#(take(N,XS)) -> isLNat#(XS)
    active#(take(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
    active#(take(N,XS)) -> isNaturalKind#(N)
    active#(take(N,XS)) -> isNatural#(N)
    active#(take(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
    active#(take(N,XS)) -> and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
    active#(take(N,XS)) ->
    U221#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
    active#(U101(X1,X2,X3)) -> active#(X1)
    active#(U101(X1,X2,X3)) -> U101#(active(X1),X2,X3)
    active#(U102(X1,X2)) -> active#(X1)
    active#(U102(X1,X2)) -> U102#(active(X1),X2)
    active#(U103(X)) -> active#(X)
    active#(U103(X)) -> U103#(active(X))
    active#(U11(X1,X2,X3)) -> active#(X1)
    active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
    active#(snd(X)) -> active#(X)
    active#(snd(X)) -> snd#(active(X))
    active#(splitAt(X1,X2)) -> active#(X1)
    active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2)
    active#(splitAt(X1,X2)) -> active#(X2)
    active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2))
    active#(U111(X1,X2)) -> active#(X1)
    active#(U111(X1,X2)) -> U111#(active(X1),X2)
    active#(U112(X)) -> active#(X)
    active#(U112(X)) -> U112#(active(X))
    active#(U121(X1,X2)) -> active#(X1)
    active#(U121(X1,X2)) -> U121#(active(X1),X2)
    active#(U122(X)) -> active#(X)
    active#(U122(X)) -> U122#(active(X))
    active#(U131(X1,X2,X3)) -> active#(X1)
    active#(U131(X1,X2,X3)) -> U131#(active(X1),X2,X3)
    active#(U132(X1,X2)) -> active#(X1)
    active#(U132(X1,X2)) -> U132#(active(X1),X2)
    active#(U133(X)) -> active#(X)
    active#(U133(X)) -> U133#(active(X))
    active#(U141(X1,X2,X3)) -> active#(X1)
    active#(U141(X1,X2,X3)) -> U141#(active(X1),X2,X3)
    active#(U142(X1,X2)) -> active#(X1)
    active#(U142(X1,X2)) -> U142#(active(X1),X2)
    active#(U143(X)) -> active#(X)
    active#(U143(X)) -> U143#(active(X))
    active#(U151(X1,X2,X3)) -> active#(X1)
    active#(U151(X1,X2,X3)) -> U151#(active(X1),X2,X3)
    active#(U152(X1,X2)) -> active#(X1)
    active#(U152(X1,X2)) -> U152#(active(X1),X2)
    active#(U153(X)) -> active#(X)
    active#(U153(X)) -> U153#(active(X))
    active#(U161(X1,X2)) -> active#(X1)
    active#(U161(X1,X2)) -> U161#(active(X1),X2)
    active#(cons(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(natsFrom(X)) -> active#(X)
    active#(natsFrom(X)) -> natsFrom#(active(X))
    active#(s(X)) -> active#(X)
    active#(s(X)) -> s#(active(X))
    active#(U171(X1,X2,X3)) -> active#(X1)
    active#(U171(X1,X2,X3)) -> U171#(active(X1),X2,X3)
    active#(head(X)) -> active#(X)
    active#(head(X)) -> head#(active(X))
    active#(afterNth(X1,X2)) -> active#(X1)
    active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2)
    active#(afterNth(X1,X2)) -> active#(X2)
    active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2))
    active#(U181(X1,X2)) -> active#(X1)
    active#(U181(X1,X2)) -> U181#(active(X1),X2)
    active#(U191(X1,X2)) -> active#(X1)
    active#(U191(X1,X2)) -> U191#(active(X1),X2)
    active#(pair(X1,X2)) -> active#(X1)
    active#(pair(X1,X2)) -> pair#(active(X1),X2)
    active#(pair(X1,X2)) -> active#(X2)
    active#(pair(X1,X2)) -> pair#(X1,active(X2))
    active#(U201(X1,X2,X3,X4)) -> active#(X1)
    active#(U201(X1,X2,X3,X4)) -> U201#(active(X1),X2,X3,X4)
    active#(U202(X1,X2)) -> active#(X1)
    active#(U202(X1,X2)) -> U202#(active(X1),X2)
    active#(U21(X1,X2)) -> active#(X1)
    active#(U21(X1,X2)) -> U21#(active(X1),X2)
    active#(U211(X1,X2)) -> active#(X1)
    active#(U211(X1,X2)) -> U211#(active(X1),X2)
    active#(U221(X1,X2,X3)) -> active#(X1)
    active#(U221(X1,X2,X3)) -> U221#(active(X1),X2,X3)
    active#(fst(X)) -> active#(X)
    active#(fst(X)) -> fst#(active(X))
    active#(U31(X1,X2)) -> active#(X1)
    active#(U31(X1,X2)) -> U31#(active(X1),X2)
    active#(U41(X1,X2,X3)) -> active#(X1)
    active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
    active#(U42(X1,X2)) -> active#(X1)
    active#(U42(X1,X2)) -> U42#(active(X1),X2)
    active#(U43(X)) -> active#(X)
    active#(U43(X)) -> U43#(active(X))
    active#(U51(X1,X2,X3)) -> active#(X1)
    active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3)
    active#(U52(X1,X2)) -> active#(X1)
    active#(U52(X1,X2)) -> U52#(active(X1),X2)
    active#(U53(X)) -> active#(X)
    active#(U53(X)) -> U53#(active(X))
    active#(U61(X1,X2)) -> active#(X1)
    active#(U61(X1,X2)) -> U61#(active(X1),X2)
    active#(U62(X)) -> active#(X)
    active#(U62(X)) -> U62#(active(X))
    active#(U71(X1,X2)) -> active#(X1)
    active#(U71(X1,X2)) -> U71#(active(X1),X2)
    active#(U72(X)) -> active#(X)
    active#(U72(X)) -> U72#(active(X))
    active#(U81(X1,X2)) -> active#(X1)
    active#(U81(X1,X2)) -> U81#(active(X1),X2)
    active#(U82(X)) -> active#(X)
    active#(U82(X)) -> U82#(active(X))
    active#(U91(X1,X2)) -> active#(X1)
    active#(U91(X1,X2)) -> U91#(active(X1),X2)
    active#(U92(X)) -> active#(X)
    active#(U92(X)) -> U92#(active(X))
    active#(and(X1,X2)) -> active#(X1)
    active#(and(X1,X2)) -> and#(active(X1),X2)
    active#(tail(X)) -> active#(X)
    active#(tail(X)) -> tail#(active(X))
    active#(take(X1,X2)) -> active#(X1)
    active#(take(X1,X2)) -> take#(active(X1),X2)
    active#(take(X1,X2)) -> active#(X2)
    active#(take(X1,X2)) -> take#(X1,active(X2))
    active#(sel(X1,X2)) -> active#(X1)
    active#(sel(X1,X2)) -> sel#(active(X1),X2)
    active#(sel(X1,X2)) -> active#(X2)
    active#(sel(X1,X2)) -> sel#(X1,active(X2))
    U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
    U102#(mark(X1),X2) -> U102#(X1,X2)
    U103#(mark(X)) -> U103#(X)
    U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
    snd#(mark(X)) -> snd#(X)
    splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
    splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
    U111#(mark(X1),X2) -> U111#(X1,X2)
    U112#(mark(X)) -> U112#(X)
    U121#(mark(X1),X2) -> U121#(X1,X2)
    U122#(mark(X)) -> U122#(X)
    U131#(mark(X1),X2,X3) -> U131#(X1,X2,X3)
    U132#(mark(X1),X2) -> U132#(X1,X2)
    U133#(mark(X)) -> U133#(X)
    U141#(mark(X1),X2,X3) -> U141#(X1,X2,X3)
    U142#(mark(X1),X2) -> U142#(X1,X2)
    U143#(mark(X)) -> U143#(X)
    U151#(mark(X1),X2,X3) -> U151#(X1,X2,X3)
    U152#(mark(X1),X2) -> U152#(X1,X2)
    U153#(mark(X)) -> U153#(X)
    U161#(mark(X1),X2) -> U161#(X1,X2)
    cons#(mark(X1),X2) -> cons#(X1,X2)
    natsFrom#(mark(X)) -> natsFrom#(X)
    s#(mark(X)) -> s#(X)
    U171#(mark(X1),X2,X3) -> U171#(X1,X2,X3)
    head#(mark(X)) -> head#(X)
    afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
    afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
    U181#(mark(X1),X2) -> U181#(X1,X2)
    U191#(mark(X1),X2) -> U191#(X1,X2)
    pair#(mark(X1),X2) -> pair#(X1,X2)
    pair#(X1,mark(X2)) -> pair#(X1,X2)
    U201#(mark(X1),X2,X3,X4) -> U201#(X1,X2,X3,X4)
    U202#(mark(X1),X2) -> U202#(X1,X2)
    U21#(mark(X1),X2) -> U21#(X1,X2)
    U211#(mark(X1),X2) -> U211#(X1,X2)
    U221#(mark(X1),X2,X3) -> U221#(X1,X2,X3)
    fst#(mark(X)) -> fst#(X)
    U31#(mark(X1),X2) -> U31#(X1,X2)
    U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
    U42#(mark(X1),X2) -> U42#(X1,X2)
    U43#(mark(X)) -> U43#(X)
    U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
    U52#(mark(X1),X2) -> U52#(X1,X2)
    U53#(mark(X)) -> U53#(X)
    U61#(mark(X1),X2) -> U61#(X1,X2)
    U62#(mark(X)) -> U62#(X)
    U71#(mark(X1),X2) -> U71#(X1,X2)
    U72#(mark(X)) -> U72#(X)
    U81#(mark(X1),X2) -> U81#(X1,X2)
    U82#(mark(X)) -> U82#(X)
    U91#(mark(X1),X2) -> U91#(X1,X2)
    U92#(mark(X)) -> U92#(X)
    and#(mark(X1),X2) -> and#(X1,X2)
    tail#(mark(X)) -> tail#(X)
    take#(mark(X1),X2) -> take#(X1,X2)
    take#(X1,mark(X2)) -> take#(X1,X2)
    sel#(mark(X1),X2) -> sel#(X1,X2)
    sel#(X1,mark(X2)) -> sel#(X1,X2)
    proper#(U101(X1,X2,X3)) -> proper#(X3)
    proper#(U101(X1,X2,X3)) -> proper#(X2)
    proper#(U101(X1,X2,X3)) -> proper#(X1)
    proper#(U101(X1,X2,X3)) -> U101#(proper(X1),proper(X2),proper(X3))
    proper#(U102(X1,X2)) -> proper#(X2)
    proper#(U102(X1,X2)) -> proper#(X1)
    proper#(U102(X1,X2)) -> U102#(proper(X1),proper(X2))
    proper#(isNatural(X)) -> proper#(X)
    proper#(isNatural(X)) -> isNatural#(proper(X))
    proper#(U103(X)) -> proper#(X)
    proper#(U103(X)) -> U103#(proper(X))
    proper#(isLNat(X)) -> proper#(X)
    proper#(isLNat(X)) -> isLNat#(proper(X))
    proper#(U11(X1,X2,X3)) -> proper#(X3)
    proper#(U11(X1,X2,X3)) -> proper#(X2)
    proper#(U11(X1,X2,X3)) -> proper#(X1)
    proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
    proper#(snd(X)) -> proper#(X)
    proper#(snd(X)) -> snd#(proper(X))
    proper#(splitAt(X1,X2)) -> proper#(X2)
    proper#(splitAt(X1,X2)) -> proper#(X1)
    proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2))
    proper#(U111(X1,X2)) -> proper#(X2)
    proper#(U111(X1,X2)) -> proper#(X1)
    proper#(U111(X1,X2)) -> U111#(proper(X1),proper(X2))
    proper#(U112(X)) -> proper#(X)
    proper#(U112(X)) -> U112#(proper(X))
    proper#(U121(X1,X2)) -> proper#(X2)
    proper#(U121(X1,X2)) -> proper#(X1)
    proper#(U121(X1,X2)) -> U121#(proper(X1),proper(X2))
    proper#(U122(X)) -> proper#(X)
    proper#(U122(X)) -> U122#(proper(X))
    proper#(U131(X1,X2,X3)) -> proper#(X3)
    proper#(U131(X1,X2,X3)) -> proper#(X2)
    proper#(U131(X1,X2,X3)) -> proper#(X1)
    proper#(U131(X1,X2,X3)) -> U131#(proper(X1),proper(X2),proper(X3))
    proper#(U132(X1,X2)) -> proper#(X2)
    proper#(U132(X1,X2)) -> proper#(X1)
    proper#(U132(X1,X2)) -> U132#(proper(X1),proper(X2))
    proper#(U133(X)) -> proper#(X)
    proper#(U133(X)) -> U133#(proper(X))
    proper#(U141(X1,X2,X3)) -> proper#(X3)
    proper#(U141(X1,X2,X3)) -> proper#(X2)
    proper#(U141(X1,X2,X3)) -> proper#(X1)
    proper#(U141(X1,X2,X3)) -> U141#(proper(X1),proper(X2),proper(X3))
    proper#(U142(X1,X2)) -> proper#(X2)
    proper#(U142(X1,X2)) -> proper#(X1)
    proper#(U142(X1,X2)) -> U142#(proper(X1),proper(X2))
    proper#(U143(X)) -> proper#(X)
    proper#(U143(X)) -> U143#(proper(X))
    proper#(U151(X1,X2,X3)) -> proper#(X3)
    proper#(U151(X1,X2,X3)) -> proper#(X2)
    proper#(U151(X1,X2,X3)) -> proper#(X1)
    proper#(U151(X1,X2,X3)) -> U151#(proper(X1),proper(X2),proper(X3))
    proper#(U152(X1,X2)) -> proper#(X2)
    proper#(U152(X1,X2)) -> proper#(X1)
    proper#(U152(X1,X2)) -> U152#(proper(X1),proper(X2))
    proper#(U153(X)) -> proper#(X)
    proper#(U153(X)) -> U153#(proper(X))
    proper#(U161(X1,X2)) -> proper#(X2)
    proper#(U161(X1,X2)) -> proper#(X1)
    proper#(U161(X1,X2)) -> U161#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(natsFrom(X)) -> proper#(X)
    proper#(natsFrom(X)) -> natsFrom#(proper(X))
    proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X))
    proper#(U171(X1,X2,X3)) -> proper#(X3)
    proper#(U171(X1,X2,X3)) -> proper#(X2)
    proper#(U171(X1,X2,X3)) -> proper#(X1)
    proper#(U171(X1,X2,X3)) -> U171#(proper(X1),proper(X2),proper(X3))
    proper#(head(X)) -> proper#(X)
    proper#(head(X)) -> head#(proper(X))
    proper#(afterNth(X1,X2)) -> proper#(X2)
    proper#(afterNth(X1,X2)) -> proper#(X1)
    proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2))
    proper#(U181(X1,X2)) -> proper#(X2)
    proper#(U181(X1,X2)) -> proper#(X1)
    proper#(U181(X1,X2)) -> U181#(proper(X1),proper(X2))
    proper#(U191(X1,X2)) -> proper#(X2)
    proper#(U191(X1,X2)) -> proper#(X1)
    proper#(U191(X1,X2)) -> U191#(proper(X1),proper(X2))
    proper#(pair(X1,X2)) -> proper#(X2)
    proper#(pair(X1,X2)) -> proper#(X1)
    proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2))
    proper#(U201(X1,X2,X3,X4)) -> proper#(X4)
    proper#(U201(X1,X2,X3,X4)) -> proper#(X3)
    proper#(U201(X1,X2,X3,X4)) -> proper#(X2)
    proper#(U201(X1,X2,X3,X4)) -> proper#(X1)
    proper#(U201(X1,X2,X3,X4)) -> U201#(proper(X1),proper(X2),proper(X3),proper(X4))
    proper#(U202(X1,X2)) -> proper#(X2)
    proper#(U202(X1,X2)) -> proper#(X1)
    proper#(U202(X1,X2)) -> U202#(proper(X1),proper(X2))
    proper#(U21(X1,X2)) -> proper#(X2)
    proper#(U21(X1,X2)) -> proper#(X1)
    proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
    proper#(U211(X1,X2)) -> proper#(X2)
    proper#(U211(X1,X2)) -> proper#(X1)
    proper#(U211(X1,X2)) -> U211#(proper(X1),proper(X2))
    proper#(U221(X1,X2,X3)) -> proper#(X3)
    proper#(U221(X1,X2,X3)) -> proper#(X2)
    proper#(U221(X1,X2,X3)) -> proper#(X1)
    proper#(U221(X1,X2,X3)) -> U221#(proper(X1),proper(X2),proper(X3))
    proper#(fst(X)) -> proper#(X)
    proper#(fst(X)) -> fst#(proper(X))
    proper#(U31(X1,X2)) -> proper#(X2)
    proper#(U31(X1,X2)) -> proper#(X1)
    proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
    proper#(U41(X1,X2,X3)) -> proper#(X3)
    proper#(U41(X1,X2,X3)) -> proper#(X2)
    proper#(U41(X1,X2,X3)) -> proper#(X1)
    proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
    proper#(U42(X1,X2)) -> proper#(X2)
    proper#(U42(X1,X2)) -> proper#(X1)
    proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2))
    proper#(U43(X)) -> proper#(X)
    proper#(U43(X)) -> U43#(proper(X))
    proper#(U51(X1,X2,X3)) -> proper#(X3)
    proper#(U51(X1,X2,X3)) -> proper#(X2)
    proper#(U51(X1,X2,X3)) -> proper#(X1)
    proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3))
    proper#(U52(X1,X2)) -> proper#(X2)
    proper#(U52(X1,X2)) -> proper#(X1)
    proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2))
    proper#(U53(X)) -> proper#(X)
    proper#(U53(X)) -> U53#(proper(X))
    proper#(U61(X1,X2)) -> proper#(X2)
    proper#(U61(X1,X2)) -> proper#(X1)
    proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2))
    proper#(U62(X)) -> proper#(X)
    proper#(U62(X)) -> U62#(proper(X))
    proper#(isPLNat(X)) -> proper#(X)
    proper#(isPLNat(X)) -> isPLNat#(proper(X))
    proper#(U71(X1,X2)) -> proper#(X2)
    proper#(U71(X1,X2)) -> proper#(X1)
    proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2))
    proper#(U72(X)) -> proper#(X)
    proper#(U72(X)) -> U72#(proper(X))
    proper#(U81(X1,X2)) -> proper#(X2)
    proper#(U81(X1,X2)) -> proper#(X1)
    proper#(U81(X1,X2)) -> U81#(proper(X1),proper(X2))
    proper#(U82(X)) -> proper#(X)
    proper#(U82(X)) -> U82#(proper(X))
    proper#(U91(X1,X2)) -> proper#(X2)
    proper#(U91(X1,X2)) -> proper#(X1)
    proper#(U91(X1,X2)) -> U91#(proper(X1),proper(X2))
    proper#(U92(X)) -> proper#(X)
    proper#(U92(X)) -> U92#(proper(X))
    proper#(and(X1,X2)) -> proper#(X2)
    proper#(and(X1,X2)) -> proper#(X1)
    proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
    proper#(isNaturalKind(X)) -> proper#(X)
    proper#(isNaturalKind(X)) -> isNaturalKind#(proper(X))
    proper#(isLNatKind(X)) -> proper#(X)
    proper#(isLNatKind(X)) -> isLNatKind#(proper(X))
    proper#(isPLNatKind(X)) -> proper#(X)
    proper#(isPLNatKind(X)) -> isPLNatKind#(proper(X))
    proper#(tail(X)) -> proper#(X)
    proper#(tail(X)) -> tail#(proper(X))
    proper#(take(X1,X2)) -> proper#(X2)
    proper#(take(X1,X2)) -> proper#(X1)
    proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
    proper#(sel(X1,X2)) -> proper#(X2)
    proper#(sel(X1,X2)) -> proper#(X1)
    proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2))
    U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3)
    U102#(ok(X1),ok(X2)) -> U102#(X1,X2)
    isNatural#(ok(X)) -> isNatural#(X)
    U103#(ok(X)) -> U103#(X)
    isLNat#(ok(X)) -> isLNat#(X)
    U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
    snd#(ok(X)) -> snd#(X)
    splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
    U111#(ok(X1),ok(X2)) -> U111#(X1,X2)
    U112#(ok(X)) -> U112#(X)
    U121#(ok(X1),ok(X2)) -> U121#(X1,X2)
    U122#(ok(X)) -> U122#(X)
    U131#(ok(X1),ok(X2),ok(X3)) -> U131#(X1,X2,X3)
    U132#(ok(X1),ok(X2)) -> U132#(X1,X2)
    U133#(ok(X)) -> U133#(X)
    U141#(ok(X1),ok(X2),ok(X3)) -> U141#(X1,X2,X3)
    U142#(ok(X1),ok(X2)) -> U142#(X1,X2)
    U143#(ok(X)) -> U143#(X)
    U151#(ok(X1),ok(X2),ok(X3)) -> U151#(X1,X2,X3)
    U152#(ok(X1),ok(X2)) -> U152#(X1,X2)
    U153#(ok(X)) -> U153#(X)
    U161#(ok(X1),ok(X2)) -> U161#(X1,X2)
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    natsFrom#(ok(X)) -> natsFrom#(X)
    s#(ok(X)) -> s#(X)
    U171#(ok(X1),ok(X2),ok(X3)) -> U171#(X1,X2,X3)
    head#(ok(X)) -> head#(X)
    afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
    U181#(ok(X1),ok(X2)) -> U181#(X1,X2)
    U191#(ok(X1),ok(X2)) -> U191#(X1,X2)
    pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
    U201#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U201#(X1,X2,X3,X4)
    U202#(ok(X1),ok(X2)) -> U202#(X1,X2)
    U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
    U211#(ok(X1),ok(X2)) -> U211#(X1,X2)
    U221#(ok(X1),ok(X2),ok(X3)) -> U221#(X1,X2,X3)
    fst#(ok(X)) -> fst#(X)
    U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
    U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
    U42#(ok(X1),ok(X2)) -> U42#(X1,X2)
    U43#(ok(X)) -> U43#(X)
    U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
    U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
    U53#(ok(X)) -> U53#(X)
    U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
    U62#(ok(X)) -> U62#(X)
    isPLNat#(ok(X)) -> isPLNat#(X)
    U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
    U72#(ok(X)) -> U72#(X)
    U81#(ok(X1),ok(X2)) -> U81#(X1,X2)
    U82#(ok(X)) -> U82#(X)
    U91#(ok(X1),ok(X2)) -> U91#(X1,X2)
    U92#(ok(X)) -> U92#(X)
    and#(ok(X1),ok(X2)) -> and#(X1,X2)
    isNaturalKind#(ok(X)) -> isNaturalKind#(X)
    isLNatKind#(ok(X)) -> isLNatKind#(X)
    isPLNatKind#(ok(X)) -> isPLNatKind#(X)
    tail#(ok(X)) -> tail#(X)
    take#(ok(X1),ok(X2)) -> take#(X1,X2)
    sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    f131(x,y) -> x
    f131(x,y) -> y
    isNatural(ok(X)) -> ok(isNatural(X))
    isLNat(ok(X)) -> ok(isLNat(X))
    splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
    splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
    splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
    s(mark(X)) -> mark(s(X))
    s(ok(X)) -> ok(s(X))
    natsFrom(mark(X)) -> mark(natsFrom(X))
    natsFrom(ok(X)) -> ok(natsFrom(X))
    afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
    afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
    afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
    cons(mark(X1),X2) -> mark(cons(X1,X2))
    cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    isPLNat(ok(X)) -> ok(isPLNat(X))
    isLNatKind(ok(X)) -> ok(isLNatKind(X))
    isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
    and(mark(X1),X2) -> mark(and(X1,X2))
    and(ok(X1),ok(X2)) -> ok(and(X1,X2))
    isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
    active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
    active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
    active(U103(tt())) -> mark(tt())
    active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
    active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
    active(U112(tt())) -> mark(tt())
    active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
    active(U122(tt())) -> mark(tt())
    active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
    active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
    active(U133(tt())) -> mark(tt())
    active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
    active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
    active(U143(tt())) -> mark(tt())
    active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
    active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
    active(U153(tt())) -> mark(tt())
    active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
    active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
    active(U181(tt(),Y)) -> mark(Y)
    active(U191(tt(),XS)) -> mark(pair(nil(),XS))
    active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
    active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
    active(U21(tt(),X)) -> mark(X)
    active(U211(tt(),XS)) -> mark(XS)
    active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
    active(U31(tt(),N)) -> mark(N)
    active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
    active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
    active(U43(tt())) -> mark(tt())
    active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
    active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
    active(U53(tt())) -> mark(tt())
    active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
    active(U62(tt())) -> mark(tt())
    active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
    active(U72(tt())) -> mark(tt())
    active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
    active(U82(tt())) -> mark(tt())
    active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
    active(U92(tt())) -> mark(tt())
    active(afterNth(N,XS)) ->
    mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
    active(and(tt(),X)) -> mark(X)
    active(fst(pair(X,Y))) ->
    mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
    active(head(cons(N,XS))) ->
    mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
    active(isLNat(nil())) -> mark(tt())
    active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
    active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
    active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
    active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
    active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
    active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
    active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
    active(isLNatKind(nil())) -> mark(tt())
    active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
    active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
    active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
    active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
    active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
    active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
    active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
    active(isNatural(0())) -> mark(tt())
    active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
    active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
    active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
    active(isNaturalKind(0())) -> mark(tt())
    active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
    active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
    active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
    active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
    active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
    active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
    active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
    active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
    active(sel(N,XS)) ->
    mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
    active(snd(pair(X,Y))) ->
    mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
    active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
    active(splitAt(s(N),cons(X,XS))) ->
    mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                         and(isLNat(XS),isLNatKind(XS)))),
              N,X,XS))
    active(tail(cons(N,XS))) ->
    mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
    active(take(N,XS)) ->
    mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
    active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
    active(U102(X1,X2)) -> U102(active(X1),X2)
    active(U103(X)) -> U103(active(X))
    active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
    active(snd(X)) -> snd(active(X))
    active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
    active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
    active(U111(X1,X2)) -> U111(active(X1),X2)
    active(U112(X)) -> U112(active(X))
    active(U121(X1,X2)) -> U121(active(X1),X2)
    active(U122(X)) -> U122(active(X))
    active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
    active(U132(X1,X2)) -> U132(active(X1),X2)
    active(U133(X)) -> U133(active(X))
    active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
    active(U142(X1,X2)) -> U142(active(X1),X2)
    active(U143(X)) -> U143(active(X))
    active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
    active(U152(X1,X2)) -> U152(active(X1),X2)
    active(U153(X)) -> U153(active(X))
    active(U161(X1,X2)) -> U161(active(X1),X2)
    active(cons(X1,X2)) -> cons(active(X1),X2)
    active(natsFrom(X)) -> natsFrom(active(X))
    active(s(X)) -> s(active(X))
    active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
    active(head(X)) -> head(active(X))
    active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
    active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
    active(U181(X1,X2)) -> U181(active(X1),X2)
    active(U191(X1,X2)) -> U191(active(X1),X2)
    active(pair(X1,X2)) -> pair(active(X1),X2)
    active(pair(X1,X2)) -> pair(X1,active(X2))
    active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
    active(U202(X1,X2)) -> U202(active(X1),X2)
    active(U21(X1,X2)) -> U21(active(X1),X2)
    active(U211(X1,X2)) -> U211(active(X1),X2)
    active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
    active(fst(X)) -> fst(active(X))
    active(U31(X1,X2)) -> U31(active(X1),X2)
    active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
    active(U42(X1,X2)) -> U42(active(X1),X2)
    active(U43(X)) -> U43(active(X))
    active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
    active(U52(X1,X2)) -> U52(active(X1),X2)
    active(U53(X)) -> U53(active(X))
    active(U61(X1,X2)) -> U61(active(X1),X2)
    active(U62(X)) -> U62(active(X))
    active(U71(X1,X2)) -> U71(active(X1),X2)
    active(U72(X)) -> U72(active(X))
    active(U81(X1,X2)) -> U81(active(X1),X2)
    active(U82(X)) -> U82(active(X))
    active(U91(X1,X2)) -> U91(active(X1),X2)
    active(U92(X)) -> U92(active(X))
    active(and(X1,X2)) -> and(active(X1),X2)
    active(tail(X)) -> tail(active(X))
    active(take(X1,X2)) -> take(active(X1),X2)
    active(take(X1,X2)) -> take(X1,active(X2))
    active(sel(X1,X2)) -> sel(active(X1),X2)
    active(sel(X1,X2)) -> sel(X1,active(X2))
    U102(mark(X1),X2) -> mark(U102(X1,X2))
    U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
    U103(mark(X)) -> mark(U103(X))
    U103(ok(X)) -> ok(U103(X))
    snd(mark(X)) -> mark(snd(X))
    snd(ok(X)) -> ok(snd(X))
    U112(mark(X)) -> mark(U112(X))
    U112(ok(X)) -> ok(U112(X))
    U122(mark(X)) -> mark(U122(X))
    U122(ok(X)) -> ok(U122(X))
    U132(mark(X1),X2) -> mark(U132(X1,X2))
    U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
    U133(mark(X)) -> mark(U133(X))
    U133(ok(X)) -> ok(U133(X))
    U142(mark(X1),X2) -> mark(U142(X1,X2))
    U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
    U143(mark(X)) -> mark(U143(X))
    U143(ok(X)) -> ok(U143(X))
    U152(mark(X1),X2) -> mark(U152(X1,X2))
    U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
    U153(mark(X)) -> mark(U153(X))
    U153(ok(X)) -> ok(U153(X))
    head(mark(X)) -> mark(head(X))
    head(ok(X)) -> ok(head(X))
    pair(X1,mark(X2)) -> mark(pair(X1,X2))
    pair(mark(X1),X2) -> mark(pair(X1,X2))
    pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
    U202(mark(X1),X2) -> mark(U202(X1,X2))
    U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
    fst(mark(X)) -> mark(fst(X))
    fst(ok(X)) -> ok(fst(X))
    U42(mark(X1),X2) -> mark(U42(X1,X2))
    U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
    U43(mark(X)) -> mark(U43(X))
    U43(ok(X)) -> ok(U43(X))
    U52(mark(X1),X2) -> mark(U52(X1,X2))
    U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
    U53(mark(X)) -> mark(U53(X))
    U53(ok(X)) -> ok(U53(X))
    U62(mark(X)) -> mark(U62(X))
    U62(ok(X)) -> ok(U62(X))
    U72(mark(X)) -> mark(U72(X))
    U72(ok(X)) -> ok(U72(X))
    U82(mark(X)) -> mark(U82(X))
    U82(ok(X)) -> ok(U82(X))
    U92(mark(X)) -> mark(U92(X))
    U92(ok(X)) -> ok(U92(X))
    U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
    U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
    U21(mark(X1),X2) -> mark(U21(X1,X2))
    U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
    U31(mark(X1),X2) -> mark(U31(X1,X2))
    U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
    U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
    U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
    U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
    U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
    U61(mark(X1),X2) -> mark(U61(X1,X2))
    U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
    U71(mark(X1),X2) -> mark(U71(X1,X2))
    U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
    U81(mark(X1),X2) -> mark(U81(X1,X2))
    U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
    U91(mark(X1),X2) -> mark(U91(X1,X2))
    U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
    U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
    U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
    U111(mark(X1),X2) -> mark(U111(X1,X2))
    U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
    U121(mark(X1),X2) -> mark(U121(X1,X2))
    U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
    U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
    U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
    U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
    U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
    U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
    U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
    U161(mark(X1),X2) -> mark(U161(X1,X2))
    U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
    U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
    U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
    U181(mark(X1),X2) -> mark(U181(X1,X2))
    U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
    U191(mark(X1),X2) -> mark(U191(X1,X2))
    U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
    U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
    U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
    U211(mark(X1),X2) -> mark(U211(X1,X2))
    U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
    U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
    U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
    tail(mark(X)) -> mark(tail(X))
    tail(ok(X)) -> ok(tail(X))
    take(mark(X1),X2) -> mark(take(X1,X2))
    take(X1,mark(X2)) -> mark(take(X1,X2))
    take(ok(X1),ok(X2)) -> ok(take(X1,X2))
    sel(mark(X1),X2) -> mark(sel(X1,X2))
    sel(X1,mark(X2)) -> mark(sel(X1,X2))
    sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
    proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
    proper(tt()) -> ok(tt())
    proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
    proper(isNatural(X)) -> isNatural(proper(X))
    proper(U103(X)) -> U103(proper(X))
    proper(isLNat(X)) -> isLNat(proper(X))
    proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
    proper(snd(X)) -> snd(proper(X))
    proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
    proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
    proper(U112(X)) -> U112(proper(X))
    proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
    proper(U122(X)) -> U122(proper(X))
    proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
    proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
    proper(U133(X)) -> U133(proper(X))
    proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
    proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
    proper(U143(X)) -> U143(proper(X))
    proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
    proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
    proper(U153(X)) -> U153(proper(X))
    proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
    proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    proper(natsFrom(X)) -> natsFrom(proper(X))
    proper(s(X)) -> s(proper(X))
    proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
    proper(head(X)) -> head(proper(X))
    proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
    proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
    proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
    proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
    proper(nil()) -> ok(nil())
    proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
    proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
    proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
    proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
    proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
    proper(fst(X)) -> fst(proper(X))
    proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
    proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
    proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
    proper(U43(X)) -> U43(proper(X))
    proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
    proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
    proper(U53(X)) -> U53(proper(X))
    proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
    proper(U62(X)) -> U62(proper(X))
    proper(isPLNat(X)) -> isPLNat(proper(X))
    proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
    proper(U72(X)) -> U72(proper(X))
    proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
    proper(U82(X)) -> U82(proper(X))
    proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
    proper(U92(X)) -> U92(proper(X))
    proper(and(X1,X2)) -> and(proper(X1),proper(X2))
    proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
    proper(isLNatKind(X)) -> isLNatKind(proper(X))
    proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
    proper(tail(X)) -> tail(proper(X))
    proper(take(X1,X2)) -> take(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
   TDG Processor:
    DPs:
     active#(U101(tt(),V1,V2)) -> isNatural#(V1)
     active#(U101(tt(),V1,V2)) -> U102#(isNatural(V1),V2)
     active#(U102(tt(),V2)) -> isLNat#(V2)
     active#(U102(tt(),V2)) -> U103#(isLNat(V2))
     active#(U11(tt(),N,XS)) -> splitAt#(N,XS)
     active#(U11(tt(),N,XS)) -> snd#(splitAt(N,XS))
     active#(U111(tt(),V1)) -> isLNat#(V1)
     active#(U111(tt(),V1)) -> U112#(isLNat(V1))
     active#(U121(tt(),V1)) -> isNatural#(V1)
     active#(U121(tt(),V1)) -> U122#(isNatural(V1))
     active#(U131(tt(),V1,V2)) -> isNatural#(V1)
     active#(U131(tt(),V1,V2)) -> U132#(isNatural(V1),V2)
     active#(U132(tt(),V2)) -> isLNat#(V2)
     active#(U132(tt(),V2)) -> U133#(isLNat(V2))
     active#(U141(tt(),V1,V2)) -> isLNat#(V1)
     active#(U141(tt(),V1,V2)) -> U142#(isLNat(V1),V2)
     active#(U142(tt(),V2)) -> isLNat#(V2)
     active#(U142(tt(),V2)) -> U143#(isLNat(V2))
     active#(U151(tt(),V1,V2)) -> isNatural#(V1)
     active#(U151(tt(),V1,V2)) -> U152#(isNatural(V1),V2)
     active#(U152(tt(),V2)) -> isLNat#(V2)
     active#(U152(tt(),V2)) -> U153#(isLNat(V2))
     active#(U161(tt(),N)) -> s#(N)
     active#(U161(tt(),N)) -> natsFrom#(s(N))
     active#(U161(tt(),N)) -> cons#(N,natsFrom(s(N)))
     active#(U171(tt(),N,XS)) -> afterNth#(N,XS)
     active#(U171(tt(),N,XS)) -> head#(afterNth(N,XS))
     active#(U191(tt(),XS)) -> pair#(nil(),XS)
     active#(U201(tt(),N,X,XS)) -> splitAt#(N,XS)
     active#(U201(tt(),N,X,XS)) -> U202#(splitAt(N,XS),X)
     active#(U202(pair(YS,ZS),X)) -> cons#(X,YS)
     active#(U202(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS)
     active#(U221(tt(),N,XS)) -> splitAt#(N,XS)
     active#(U221(tt(),N,XS)) -> fst#(splitAt(N,XS))
     active#(U41(tt(),V1,V2)) -> isNatural#(V1)
     active#(U41(tt(),V1,V2)) -> U42#(isNatural(V1),V2)
     active#(U42(tt(),V2)) -> isLNat#(V2)
     active#(U42(tt(),V2)) -> U43#(isLNat(V2))
     active#(U51(tt(),V1,V2)) -> isNatural#(V1)
     active#(U51(tt(),V1,V2)) -> U52#(isNatural(V1),V2)
     active#(U52(tt(),V2)) -> isLNat#(V2)
     active#(U52(tt(),V2)) -> U53#(isLNat(V2))
     active#(U61(tt(),V1)) -> isPLNat#(V1)
     active#(U61(tt(),V1)) -> U62#(isPLNat(V1))
     active#(U71(tt(),V1)) -> isNatural#(V1)
     active#(U71(tt(),V1)) -> U72#(isNatural(V1))
     active#(U81(tt(),V1)) -> isPLNat#(V1)
     active#(U81(tt(),V1)) -> U82#(isPLNat(V1))
     active#(U91(tt(),V1)) -> isLNat#(V1)
     active#(U91(tt(),V1)) -> U92#(isLNat(V1))
     active#(afterNth(N,XS)) -> isLNatKind#(XS)
     active#(afterNth(N,XS)) -> isLNat#(XS)
     active#(afterNth(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
     active#(afterNth(N,XS)) -> isNaturalKind#(N)
     active#(afterNth(N,XS)) -> isNatural#(N)
     active#(afterNth(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
     active#(afterNth(N,XS)) ->
     and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
     active#(afterNth(N,XS)) ->
     U11#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
     active#(fst(pair(X,Y))) -> isLNatKind#(Y)
     active#(fst(pair(X,Y))) -> isLNat#(Y)
     active#(fst(pair(X,Y))) -> and#(isLNat(Y),isLNatKind(Y))
     active#(fst(pair(X,Y))) -> isLNatKind#(X)
     active#(fst(pair(X,Y))) -> isLNat#(X)
     active#(fst(pair(X,Y))) -> and#(isLNat(X),isLNatKind(X))
     active#(fst(pair(X,Y))) -> and#(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y)))
     active#(fst(pair(X,Y))) ->
     U21#(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X)
     active#(head(cons(N,XS))) -> isLNatKind#(XS)
     active#(head(cons(N,XS))) -> isLNat#(XS)
     active#(head(cons(N,XS))) -> and#(isLNat(XS),isLNatKind(XS))
     active#(head(cons(N,XS))) -> isNaturalKind#(N)
     active#(head(cons(N,XS))) -> isNatural#(N)
     active#(head(cons(N,XS))) -> and#(isNatural(N),isNaturalKind(N))
     active#(head(cons(N,XS))) ->
     and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
     active#(head(cons(N,XS))) ->
     U31#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N)
     active#(isLNat(afterNth(V1,V2))) -> isLNatKind#(V2)
     active#(isLNat(afterNth(V1,V2))) -> isNaturalKind#(V1)
     active#(isLNat(afterNth(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isLNat(afterNth(V1,V2))) -> U41#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
     active#(isLNat(cons(V1,V2))) -> isLNatKind#(V2)
     active#(isLNat(cons(V1,V2))) -> isNaturalKind#(V1)
     active#(isLNat(cons(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isLNat(cons(V1,V2))) -> U51#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
     active#(isLNat(fst(V1))) -> isPLNatKind#(V1)
     active#(isLNat(fst(V1))) -> U61#(isPLNatKind(V1),V1)
     active#(isLNat(natsFrom(V1))) -> isNaturalKind#(V1)
     active#(isLNat(natsFrom(V1))) -> U71#(isNaturalKind(V1),V1)
     active#(isLNat(snd(V1))) -> isPLNatKind#(V1)
     active#(isLNat(snd(V1))) -> U81#(isPLNatKind(V1),V1)
     active#(isLNat(tail(V1))) -> isLNatKind#(V1)
     active#(isLNat(tail(V1))) -> U91#(isLNatKind(V1),V1)
     active#(isLNat(take(V1,V2))) -> isLNatKind#(V2)
     active#(isLNat(take(V1,V2))) -> isNaturalKind#(V1)
     active#(isLNat(take(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isLNat(take(V1,V2))) -> U101#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
     active#(isLNatKind(afterNth(V1,V2))) -> isLNatKind#(V2)
     active#(isLNatKind(afterNth(V1,V2))) -> isNaturalKind#(V1)
     active#(isLNatKind(afterNth(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isLNatKind(cons(V1,V2))) -> isLNatKind#(V2)
     active#(isLNatKind(cons(V1,V2))) -> isNaturalKind#(V1)
     active#(isLNatKind(cons(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isLNatKind(fst(V1))) -> isPLNatKind#(V1)
     active#(isLNatKind(natsFrom(V1))) -> isNaturalKind#(V1)
     active#(isLNatKind(snd(V1))) -> isPLNatKind#(V1)
     active#(isLNatKind(tail(V1))) -> isLNatKind#(V1)
     active#(isLNatKind(take(V1,V2))) -> isLNatKind#(V2)
     active#(isLNatKind(take(V1,V2))) -> isNaturalKind#(V1)
     active#(isLNatKind(take(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isNatural(head(V1))) -> isLNatKind#(V1)
     active#(isNatural(head(V1))) -> U111#(isLNatKind(V1),V1)
     active#(isNatural(s(V1))) -> isNaturalKind#(V1)
     active#(isNatural(s(V1))) -> U121#(isNaturalKind(V1),V1)
     active#(isNatural(sel(V1,V2))) -> isLNatKind#(V2)
     active#(isNatural(sel(V1,V2))) -> isNaturalKind#(V1)
     active#(isNatural(sel(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isNatural(sel(V1,V2))) -> U131#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
     active#(isNaturalKind(head(V1))) -> isLNatKind#(V1)
     active#(isNaturalKind(s(V1))) -> isNaturalKind#(V1)
     active#(isNaturalKind(sel(V1,V2))) -> isLNatKind#(V2)
     active#(isNaturalKind(sel(V1,V2))) -> isNaturalKind#(V1)
     active#(isNaturalKind(sel(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isPLNat(pair(V1,V2))) -> isLNatKind#(V2)
     active#(isPLNat(pair(V1,V2))) -> isLNatKind#(V1)
     active#(isPLNat(pair(V1,V2))) -> and#(isLNatKind(V1),isLNatKind(V2))
     active#(isPLNat(pair(V1,V2))) -> U141#(and(isLNatKind(V1),isLNatKind(V2)),V1,V2)
     active#(isPLNat(splitAt(V1,V2))) -> isLNatKind#(V2)
     active#(isPLNat(splitAt(V1,V2))) -> isNaturalKind#(V1)
     active#(isPLNat(splitAt(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(isPLNat(splitAt(V1,V2))) -> U151#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
     active#(isPLNatKind(pair(V1,V2))) -> isLNatKind#(V2)
     active#(isPLNatKind(pair(V1,V2))) -> isLNatKind#(V1)
     active#(isPLNatKind(pair(V1,V2))) -> and#(isLNatKind(V1),isLNatKind(V2))
     active#(isPLNatKind(splitAt(V1,V2))) -> isLNatKind#(V2)
     active#(isPLNatKind(splitAt(V1,V2))) -> isNaturalKind#(V1)
     active#(isPLNatKind(splitAt(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
     active#(natsFrom(N)) -> isNaturalKind#(N)
     active#(natsFrom(N)) -> isNatural#(N)
     active#(natsFrom(N)) -> and#(isNatural(N),isNaturalKind(N))
     active#(natsFrom(N)) -> U161#(and(isNatural(N),isNaturalKind(N)),N)
     active#(sel(N,XS)) -> isLNatKind#(XS)
     active#(sel(N,XS)) -> isLNat#(XS)
     active#(sel(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
     active#(sel(N,XS)) -> isNaturalKind#(N)
     active#(sel(N,XS)) -> isNatural#(N)
     active#(sel(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
     active#(sel(N,XS)) -> and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
     active#(sel(N,XS)) ->
     U171#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
     active#(snd(pair(X,Y))) -> isLNatKind#(Y)
     active#(snd(pair(X,Y))) -> isLNat#(Y)
     active#(snd(pair(X,Y))) -> and#(isLNat(Y),isLNatKind(Y))
     active#(snd(pair(X,Y))) -> isLNatKind#(X)
     active#(snd(pair(X,Y))) -> isLNat#(X)
     active#(snd(pair(X,Y))) -> and#(isLNat(X),isLNatKind(X))
     active#(snd(pair(X,Y))) -> and#(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y)))
     active#(snd(pair(X,Y))) ->
     U181#(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y)
     active#(splitAt(0(),XS)) -> isLNatKind#(XS)
     active#(splitAt(0(),XS)) -> isLNat#(XS)
     active#(splitAt(0(),XS)) -> and#(isLNat(XS),isLNatKind(XS))
     active#(splitAt(0(),XS)) -> U191#(and(isLNat(XS),isLNatKind(XS)),XS)
     active#(splitAt(s(N),cons(X,XS))) -> isLNatKind#(XS)
     active#(splitAt(s(N),cons(X,XS))) -> isLNat#(XS)
     active#(splitAt(s(N),cons(X,XS))) -> and#(isLNat(XS),isLNatKind(XS))
     active#(splitAt(s(N),cons(X,XS))) -> isNaturalKind#(X)
     active#(splitAt(s(N),cons(X,XS))) -> isNatural#(X)
     active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(X),isNaturalKind(X))
     active#(splitAt(s(N),cons(X,XS))) ->
     and#(and(isNatural(X),isNaturalKind(X)),and(isLNat(XS),isLNatKind(XS)))
     active#(splitAt(s(N),cons(X,XS))) -> isNaturalKind#(N)
     active#(splitAt(s(N),cons(X,XS))) -> isNatural#(N)
     active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(N),isNaturalKind(N))
     active#(splitAt(s(N),cons(X,XS))) ->
     and#(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                 and(isLNat(XS),isLNatKind(XS))))
     active#(splitAt(s(N),cons(X,XS))) ->
     U201#(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                      and(isLNat(XS),isLNatKind(XS)))),
           N,X,XS)
     active#(tail(cons(N,XS))) -> isLNatKind#(XS)
     active#(tail(cons(N,XS))) -> isLNat#(XS)
     active#(tail(cons(N,XS))) -> and#(isLNat(XS),isLNatKind(XS))
     active#(tail(cons(N,XS))) -> isNaturalKind#(N)
     active#(tail(cons(N,XS))) -> isNatural#(N)
     active#(tail(cons(N,XS))) -> and#(isNatural(N),isNaturalKind(N))
     active#(tail(cons(N,XS))) ->
     and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
     active#(tail(cons(N,XS))) ->
     U211#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS)
     active#(take(N,XS)) -> isLNatKind#(XS)
     active#(take(N,XS)) -> isLNat#(XS)
     active#(take(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
     active#(take(N,XS)) -> isNaturalKind#(N)
     active#(take(N,XS)) -> isNatural#(N)
     active#(take(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
     active#(take(N,XS)) -> and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
     active#(take(N,XS)) ->
     U221#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
     active#(U101(X1,X2,X3)) -> active#(X1)
     active#(U101(X1,X2,X3)) -> U101#(active(X1),X2,X3)
     active#(U102(X1,X2)) -> active#(X1)
     active#(U102(X1,X2)) -> U102#(active(X1),X2)
     active#(U103(X)) -> active#(X)
     active#(U103(X)) -> U103#(active(X))
     active#(U11(X1,X2,X3)) -> active#(X1)
     active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
     active#(snd(X)) -> active#(X)
     active#(snd(X)) -> snd#(active(X))
     active#(splitAt(X1,X2)) -> active#(X1)
     active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2)
     active#(splitAt(X1,X2)) -> active#(X2)
     active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2))
     active#(U111(X1,X2)) -> active#(X1)
     active#(U111(X1,X2)) -> U111#(active(X1),X2)
     active#(U112(X)) -> active#(X)
     active#(U112(X)) -> U112#(active(X))
     active#(U121(X1,X2)) -> active#(X1)
     active#(U121(X1,X2)) -> U121#(active(X1),X2)
     active#(U122(X)) -> active#(X)
     active#(U122(X)) -> U122#(active(X))
     active#(U131(X1,X2,X3)) -> active#(X1)
     active#(U131(X1,X2,X3)) -> U131#(active(X1),X2,X3)
     active#(U132(X1,X2)) -> active#(X1)
     active#(U132(X1,X2)) -> U132#(active(X1),X2)
     active#(U133(X)) -> active#(X)
     active#(U133(X)) -> U133#(active(X))
     active#(U141(X1,X2,X3)) -> active#(X1)
     active#(U141(X1,X2,X3)) -> U141#(active(X1),X2,X3)
     active#(U142(X1,X2)) -> active#(X1)
     active#(U142(X1,X2)) -> U142#(active(X1),X2)
     active#(U143(X)) -> active#(X)
     active#(U143(X)) -> U143#(active(X))
     active#(U151(X1,X2,X3)) -> active#(X1)
     active#(U151(X1,X2,X3)) -> U151#(active(X1),X2,X3)
     active#(U152(X1,X2)) -> active#(X1)
     active#(U152(X1,X2)) -> U152#(active(X1),X2)
     active#(U153(X)) -> active#(X)
     active#(U153(X)) -> U153#(active(X))
     active#(U161(X1,X2)) -> active#(X1)
     active#(U161(X1,X2)) -> U161#(active(X1),X2)
     active#(cons(X1,X2)) -> active#(X1)
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(natsFrom(X)) -> active#(X)
     active#(natsFrom(X)) -> natsFrom#(active(X))
     active#(s(X)) -> active#(X)
     active#(s(X)) -> s#(active(X))
     active#(U171(X1,X2,X3)) -> active#(X1)
     active#(U171(X1,X2,X3)) -> U171#(active(X1),X2,X3)
     active#(head(X)) -> active#(X)
     active#(head(X)) -> head#(active(X))
     active#(afterNth(X1,X2)) -> active#(X1)
     active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2)
     active#(afterNth(X1,X2)) -> active#(X2)
     active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2))
     active#(U181(X1,X2)) -> active#(X1)
     active#(U181(X1,X2)) -> U181#(active(X1),X2)
     active#(U191(X1,X2)) -> active#(X1)
     active#(U191(X1,X2)) -> U191#(active(X1),X2)
     active#(pair(X1,X2)) -> active#(X1)
     active#(pair(X1,X2)) -> pair#(active(X1),X2)
     active#(pair(X1,X2)) -> active#(X2)
     active#(pair(X1,X2)) -> pair#(X1,active(X2))
     active#(U201(X1,X2,X3,X4)) -> active#(X1)
     active#(U201(X1,X2,X3,X4)) -> U201#(active(X1),X2,X3,X4)
     active#(U202(X1,X2)) -> active#(X1)
     active#(U202(X1,X2)) -> U202#(active(X1),X2)
     active#(U21(X1,X2)) -> active#(X1)
     active#(U21(X1,X2)) -> U21#(active(X1),X2)
     active#(U211(X1,X2)) -> active#(X1)
     active#(U211(X1,X2)) -> U211#(active(X1),X2)
     active#(U221(X1,X2,X3)) -> active#(X1)
     active#(U221(X1,X2,X3)) -> U221#(active(X1),X2,X3)
     active#(fst(X)) -> active#(X)
     active#(fst(X)) -> fst#(active(X))
     active#(U31(X1,X2)) -> active#(X1)
     active#(U31(X1,X2)) -> U31#(active(X1),X2)
     active#(U41(X1,X2,X3)) -> active#(X1)
     active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
     active#(U42(X1,X2)) -> active#(X1)
     active#(U42(X1,X2)) -> U42#(active(X1),X2)
     active#(U43(X)) -> active#(X)
     active#(U43(X)) -> U43#(active(X))
     active#(U51(X1,X2,X3)) -> active#(X1)
     active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3)
     active#(U52(X1,X2)) -> active#(X1)
     active#(U52(X1,X2)) -> U52#(active(X1),X2)
     active#(U53(X)) -> active#(X)
     active#(U53(X)) -> U53#(active(X))
     active#(U61(X1,X2)) -> active#(X1)
     active#(U61(X1,X2)) -> U61#(active(X1),X2)
     active#(U62(X)) -> active#(X)
     active#(U62(X)) -> U62#(active(X))
     active#(U71(X1,X2)) -> active#(X1)
     active#(U71(X1,X2)) -> U71#(active(X1),X2)
     active#(U72(X)) -> active#(X)
     active#(U72(X)) -> U72#(active(X))
     active#(U81(X1,X2)) -> active#(X1)
     active#(U81(X1,X2)) -> U81#(active(X1),X2)
     active#(U82(X)) -> active#(X)
     active#(U82(X)) -> U82#(active(X))
     active#(U91(X1,X2)) -> active#(X1)
     active#(U91(X1,X2)) -> U91#(active(X1),X2)
     active#(U92(X)) -> active#(X)
     active#(U92(X)) -> U92#(active(X))
     active#(and(X1,X2)) -> active#(X1)
     active#(and(X1,X2)) -> and#(active(X1),X2)
     active#(tail(X)) -> active#(X)
     active#(tail(X)) -> tail#(active(X))
     active#(take(X1,X2)) -> active#(X1)
     active#(take(X1,X2)) -> take#(active(X1),X2)
     active#(take(X1,X2)) -> active#(X2)
     active#(take(X1,X2)) -> take#(X1,active(X2))
     active#(sel(X1,X2)) -> active#(X1)
     active#(sel(X1,X2)) -> sel#(active(X1),X2)
     active#(sel(X1,X2)) -> active#(X2)
     active#(sel(X1,X2)) -> sel#(X1,active(X2))
     U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
     U102#(mark(X1),X2) -> U102#(X1,X2)
     U103#(mark(X)) -> U103#(X)
     U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
     snd#(mark(X)) -> snd#(X)
     splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
     splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
     U111#(mark(X1),X2) -> U111#(X1,X2)
     U112#(mark(X)) -> U112#(X)
     U121#(mark(X1),X2) -> U121#(X1,X2)
     U122#(mark(X)) -> U122#(X)
     U131#(mark(X1),X2,X3) -> U131#(X1,X2,X3)
     U132#(mark(X1),X2) -> U132#(X1,X2)
     U133#(mark(X)) -> U133#(X)
     U141#(mark(X1),X2,X3) -> U141#(X1,X2,X3)
     U142#(mark(X1),X2) -> U142#(X1,X2)
     U143#(mark(X)) -> U143#(X)
     U151#(mark(X1),X2,X3) -> U151#(X1,X2,X3)
     U152#(mark(X1),X2) -> U152#(X1,X2)
     U153#(mark(X)) -> U153#(X)
     U161#(mark(X1),X2) -> U161#(X1,X2)
     cons#(mark(X1),X2) -> cons#(X1,X2)
     natsFrom#(mark(X)) -> natsFrom#(X)
     s#(mark(X)) -> s#(X)
     U171#(mark(X1),X2,X3) -> U171#(X1,X2,X3)
     head#(mark(X)) -> head#(X)
     afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
     afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
     U181#(mark(X1),X2) -> U181#(X1,X2)
     U191#(mark(X1),X2) -> U191#(X1,X2)
     pair#(mark(X1),X2) -> pair#(X1,X2)
     pair#(X1,mark(X2)) -> pair#(X1,X2)
     U201#(mark(X1),X2,X3,X4) -> U201#(X1,X2,X3,X4)
     U202#(mark(X1),X2) -> U202#(X1,X2)
     U21#(mark(X1),X2) -> U21#(X1,X2)
     U211#(mark(X1),X2) -> U211#(X1,X2)
     U221#(mark(X1),X2,X3) -> U221#(X1,X2,X3)
     fst#(mark(X)) -> fst#(X)
     U31#(mark(X1),X2) -> U31#(X1,X2)
     U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
     U42#(mark(X1),X2) -> U42#(X1,X2)
     U43#(mark(X)) -> U43#(X)
     U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
     U52#(mark(X1),X2) -> U52#(X1,X2)
     U53#(mark(X)) -> U53#(X)
     U61#(mark(X1),X2) -> U61#(X1,X2)
     U62#(mark(X)) -> U62#(X)
     U71#(mark(X1),X2) -> U71#(X1,X2)
     U72#(mark(X)) -> U72#(X)
     U81#(mark(X1),X2) -> U81#(X1,X2)
     U82#(mark(X)) -> U82#(X)
     U91#(mark(X1),X2) -> U91#(X1,X2)
     U92#(mark(X)) -> U92#(X)
     and#(mark(X1),X2) -> and#(X1,X2)
     tail#(mark(X)) -> tail#(X)
     take#(mark(X1),X2) -> take#(X1,X2)
     take#(X1,mark(X2)) -> take#(X1,X2)
     sel#(mark(X1),X2) -> sel#(X1,X2)
     sel#(X1,mark(X2)) -> sel#(X1,X2)
     proper#(U101(X1,X2,X3)) -> proper#(X3)
     proper#(U101(X1,X2,X3)) -> proper#(X2)
     proper#(U101(X1,X2,X3)) -> proper#(X1)
     proper#(U101(X1,X2,X3)) -> U101#(proper(X1),proper(X2),proper(X3))
     proper#(U102(X1,X2)) -> proper#(X2)
     proper#(U102(X1,X2)) -> proper#(X1)
     proper#(U102(X1,X2)) -> U102#(proper(X1),proper(X2))
     proper#(isNatural(X)) -> proper#(X)
     proper#(isNatural(X)) -> isNatural#(proper(X))
     proper#(U103(X)) -> proper#(X)
     proper#(U103(X)) -> U103#(proper(X))
     proper#(isLNat(X)) -> proper#(X)
     proper#(isLNat(X)) -> isLNat#(proper(X))
     proper#(U11(X1,X2,X3)) -> proper#(X3)
     proper#(U11(X1,X2,X3)) -> proper#(X2)
     proper#(U11(X1,X2,X3)) -> proper#(X1)
     proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
     proper#(snd(X)) -> proper#(X)
     proper#(snd(X)) -> snd#(proper(X))
     proper#(splitAt(X1,X2)) -> proper#(X2)
     proper#(splitAt(X1,X2)) -> proper#(X1)
     proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2))
     proper#(U111(X1,X2)) -> proper#(X2)
     proper#(U111(X1,X2)) -> proper#(X1)
     proper#(U111(X1,X2)) -> U111#(proper(X1),proper(X2))
     proper#(U112(X)) -> proper#(X)
     proper#(U112(X)) -> U112#(proper(X))
     proper#(U121(X1,X2)) -> proper#(X2)
     proper#(U121(X1,X2)) -> proper#(X1)
     proper#(U121(X1,X2)) -> U121#(proper(X1),proper(X2))
     proper#(U122(X)) -> proper#(X)
     proper#(U122(X)) -> U122#(proper(X))
     proper#(U131(X1,X2,X3)) -> proper#(X3)
     proper#(U131(X1,X2,X3)) -> proper#(X2)
     proper#(U131(X1,X2,X3)) -> proper#(X1)
     proper#(U131(X1,X2,X3)) -> U131#(proper(X1),proper(X2),proper(X3))
     proper#(U132(X1,X2)) -> proper#(X2)
     proper#(U132(X1,X2)) -> proper#(X1)
     proper#(U132(X1,X2)) -> U132#(proper(X1),proper(X2))
     proper#(U133(X)) -> proper#(X)
     proper#(U133(X)) -> U133#(proper(X))
     proper#(U141(X1,X2,X3)) -> proper#(X3)
     proper#(U141(X1,X2,X3)) -> proper#(X2)
     proper#(U141(X1,X2,X3)) -> proper#(X1)
     proper#(U141(X1,X2,X3)) -> U141#(proper(X1),proper(X2),proper(X3))
     proper#(U142(X1,X2)) -> proper#(X2)
     proper#(U142(X1,X2)) -> proper#(X1)
     proper#(U142(X1,X2)) -> U142#(proper(X1),proper(X2))
     proper#(U143(X)) -> proper#(X)
     proper#(U143(X)) -> U143#(proper(X))
     proper#(U151(X1,X2,X3)) -> proper#(X3)
     proper#(U151(X1,X2,X3)) -> proper#(X2)
     proper#(U151(X1,X2,X3)) -> proper#(X1)
     proper#(U151(X1,X2,X3)) -> U151#(proper(X1),proper(X2),proper(X3))
     proper#(U152(X1,X2)) -> proper#(X2)
     proper#(U152(X1,X2)) -> proper#(X1)
     proper#(U152(X1,X2)) -> U152#(proper(X1),proper(X2))
     proper#(U153(X)) -> proper#(X)
     proper#(U153(X)) -> U153#(proper(X))
     proper#(U161(X1,X2)) -> proper#(X2)
     proper#(U161(X1,X2)) -> proper#(X1)
     proper#(U161(X1,X2)) -> U161#(proper(X1),proper(X2))
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(natsFrom(X)) -> proper#(X)
     proper#(natsFrom(X)) -> natsFrom#(proper(X))
     proper#(s(X)) -> proper#(X)
     proper#(s(X)) -> s#(proper(X))
     proper#(U171(X1,X2,X3)) -> proper#(X3)
     proper#(U171(X1,X2,X3)) -> proper#(X2)
     proper#(U171(X1,X2,X3)) -> proper#(X1)
     proper#(U171(X1,X2,X3)) -> U171#(proper(X1),proper(X2),proper(X3))
     proper#(head(X)) -> proper#(X)
     proper#(head(X)) -> head#(proper(X))
     proper#(afterNth(X1,X2)) -> proper#(X2)
     proper#(afterNth(X1,X2)) -> proper#(X1)
     proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2))
     proper#(U181(X1,X2)) -> proper#(X2)
     proper#(U181(X1,X2)) -> proper#(X1)
     proper#(U181(X1,X2)) -> U181#(proper(X1),proper(X2))
     proper#(U191(X1,X2)) -> proper#(X2)
     proper#(U191(X1,X2)) -> proper#(X1)
     proper#(U191(X1,X2)) -> U191#(proper(X1),proper(X2))
     proper#(pair(X1,X2)) -> proper#(X2)
     proper#(pair(X1,X2)) -> proper#(X1)
     proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2))
     proper#(U201(X1,X2,X3,X4)) -> proper#(X4)
     proper#(U201(X1,X2,X3,X4)) -> proper#(X3)
     proper#(U201(X1,X2,X3,X4)) -> proper#(X2)
     proper#(U201(X1,X2,X3,X4)) -> proper#(X1)
     proper#(U201(X1,X2,X3,X4)) -> U201#(proper(X1),proper(X2),proper(X3),proper(X4))
     proper#(U202(X1,X2)) -> proper#(X2)
     proper#(U202(X1,X2)) -> proper#(X1)
     proper#(U202(X1,X2)) -> U202#(proper(X1),proper(X2))
     proper#(U21(X1,X2)) -> proper#(X2)
     proper#(U21(X1,X2)) -> proper#(X1)
     proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
     proper#(U211(X1,X2)) -> proper#(X2)
     proper#(U211(X1,X2)) -> proper#(X1)
     proper#(U211(X1,X2)) -> U211#(proper(X1),proper(X2))
     proper#(U221(X1,X2,X3)) -> proper#(X3)
     proper#(U221(X1,X2,X3)) -> proper#(X2)
     proper#(U221(X1,X2,X3)) -> proper#(X1)
     proper#(U221(X1,X2,X3)) -> U221#(proper(X1),proper(X2),proper(X3))
     proper#(fst(X)) -> proper#(X)
     proper#(fst(X)) -> fst#(proper(X))
     proper#(U31(X1,X2)) -> proper#(X2)
     proper#(U31(X1,X2)) -> proper#(X1)
     proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
     proper#(U41(X1,X2,X3)) -> proper#(X3)
     proper#(U41(X1,X2,X3)) -> proper#(X2)
     proper#(U41(X1,X2,X3)) -> proper#(X1)
     proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
     proper#(U42(X1,X2)) -> proper#(X2)
     proper#(U42(X1,X2)) -> proper#(X1)
     proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2))
     proper#(U43(X)) -> proper#(X)
     proper#(U43(X)) -> U43#(proper(X))
     proper#(U51(X1,X2,X3)) -> proper#(X3)
     proper#(U51(X1,X2,X3)) -> proper#(X2)
     proper#(U51(X1,X2,X3)) -> proper#(X1)
     proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3))
     proper#(U52(X1,X2)) -> proper#(X2)
     proper#(U52(X1,X2)) -> proper#(X1)
     proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2))
     proper#(U53(X)) -> proper#(X)
     proper#(U53(X)) -> U53#(proper(X))
     proper#(U61(X1,X2)) -> proper#(X2)
     proper#(U61(X1,X2)) -> proper#(X1)
     proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2))
     proper#(U62(X)) -> proper#(X)
     proper#(U62(X)) -> U62#(proper(X))
     proper#(isPLNat(X)) -> proper#(X)
     proper#(isPLNat(X)) -> isPLNat#(proper(X))
     proper#(U71(X1,X2)) -> proper#(X2)
     proper#(U71(X1,X2)) -> proper#(X1)
     proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2))
     proper#(U72(X)) -> proper#(X)
     proper#(U72(X)) -> U72#(proper(X))
     proper#(U81(X1,X2)) -> proper#(X2)
     proper#(U81(X1,X2)) -> proper#(X1)
     proper#(U81(X1,X2)) -> U81#(proper(X1),proper(X2))
     proper#(U82(X)) -> proper#(X)
     proper#(U82(X)) -> U82#(proper(X))
     proper#(U91(X1,X2)) -> proper#(X2)
     proper#(U91(X1,X2)) -> proper#(X1)
     proper#(U91(X1,X2)) -> U91#(proper(X1),proper(X2))
     proper#(U92(X)) -> proper#(X)
     proper#(U92(X)) -> U92#(proper(X))
     proper#(and(X1,X2)) -> proper#(X2)
     proper#(and(X1,X2)) -> proper#(X1)
     proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
     proper#(isNaturalKind(X)) -> proper#(X)
     proper#(isNaturalKind(X)) -> isNaturalKind#(proper(X))
     proper#(isLNatKind(X)) -> proper#(X)
     proper#(isLNatKind(X)) -> isLNatKind#(proper(X))
     proper#(isPLNatKind(X)) -> proper#(X)
     proper#(isPLNatKind(X)) -> isPLNatKind#(proper(X))
     proper#(tail(X)) -> proper#(X)
     proper#(tail(X)) -> tail#(proper(X))
     proper#(take(X1,X2)) -> proper#(X2)
     proper#(take(X1,X2)) -> proper#(X1)
     proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
     proper#(sel(X1,X2)) -> proper#(X2)
     proper#(sel(X1,X2)) -> proper#(X1)
     proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2))
     U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3)
     U102#(ok(X1),ok(X2)) -> U102#(X1,X2)
     isNatural#(ok(X)) -> isNatural#(X)
     U103#(ok(X)) -> U103#(X)
     isLNat#(ok(X)) -> isLNat#(X)
     U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
     snd#(ok(X)) -> snd#(X)
     splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
     U111#(ok(X1),ok(X2)) -> U111#(X1,X2)
     U112#(ok(X)) -> U112#(X)
     U121#(ok(X1),ok(X2)) -> U121#(X1,X2)
     U122#(ok(X)) -> U122#(X)
     U131#(ok(X1),ok(X2),ok(X3)) -> U131#(X1,X2,X3)
     U132#(ok(X1),ok(X2)) -> U132#(X1,X2)
     U133#(ok(X)) -> U133#(X)
     U141#(ok(X1),ok(X2),ok(X3)) -> U141#(X1,X2,X3)
     U142#(ok(X1),ok(X2)) -> U142#(X1,X2)
     U143#(ok(X)) -> U143#(X)
     U151#(ok(X1),ok(X2),ok(X3)) -> U151#(X1,X2,X3)
     U152#(ok(X1),ok(X2)) -> U152#(X1,X2)
     U153#(ok(X)) -> U153#(X)
     U161#(ok(X1),ok(X2)) -> U161#(X1,X2)
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     natsFrom#(ok(X)) -> natsFrom#(X)
     s#(ok(X)) -> s#(X)
     U171#(ok(X1),ok(X2),ok(X3)) -> U171#(X1,X2,X3)
     head#(ok(X)) -> head#(X)
     afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
     U181#(ok(X1),ok(X2)) -> U181#(X1,X2)
     U191#(ok(X1),ok(X2)) -> U191#(X1,X2)
     pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
     U201#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U201#(X1,X2,X3,X4)
     U202#(ok(X1),ok(X2)) -> U202#(X1,X2)
     U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
     U211#(ok(X1),ok(X2)) -> U211#(X1,X2)
     U221#(ok(X1),ok(X2),ok(X3)) -> U221#(X1,X2,X3)
     fst#(ok(X)) -> fst#(X)
     U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
     U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
     U42#(ok(X1),ok(X2)) -> U42#(X1,X2)
     U43#(ok(X)) -> U43#(X)
     U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
     U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
     U53#(ok(X)) -> U53#(X)
     U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
     U62#(ok(X)) -> U62#(X)
     isPLNat#(ok(X)) -> isPLNat#(X)
     U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
     U72#(ok(X)) -> U72#(X)
     U81#(ok(X1),ok(X2)) -> U81#(X1,X2)
     U82#(ok(X)) -> U82#(X)
     U91#(ok(X1),ok(X2)) -> U91#(X1,X2)
     U92#(ok(X)) -> U92#(X)
     and#(ok(X1),ok(X2)) -> and#(X1,X2)
     isNaturalKind#(ok(X)) -> isNaturalKind#(X)
     isLNatKind#(ok(X)) -> isLNatKind#(X)
     isPLNatKind#(ok(X)) -> isPLNatKind#(X)
     tail#(ok(X)) -> tail#(X)
     take#(ok(X1),ok(X2)) -> take#(X1,X2)
     sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
     top#(mark(X)) -> proper#(X)
     top#(mark(X)) -> top#(proper(X))
     top#(ok(X)) -> active#(X)
     top#(ok(X)) -> top#(active(X))
    TRS:
     f131(x,y) -> x
     f131(x,y) -> y
     isNatural(ok(X)) -> ok(isNatural(X))
     isLNat(ok(X)) -> ok(isLNat(X))
     splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
     splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
     splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
     s(mark(X)) -> mark(s(X))
     s(ok(X)) -> ok(s(X))
     natsFrom(mark(X)) -> mark(natsFrom(X))
     natsFrom(ok(X)) -> ok(natsFrom(X))
     afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
     afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
     afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     isPLNat(ok(X)) -> ok(isPLNat(X))
     isLNatKind(ok(X)) -> ok(isLNatKind(X))
     isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
     and(mark(X1),X2) -> mark(and(X1,X2))
     and(ok(X1),ok(X2)) -> ok(and(X1,X2))
     isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
     active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
     active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
     active(U103(tt())) -> mark(tt())
     active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
     active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
     active(U112(tt())) -> mark(tt())
     active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
     active(U122(tt())) -> mark(tt())
     active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
     active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
     active(U133(tt())) -> mark(tt())
     active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
     active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
     active(U143(tt())) -> mark(tt())
     active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
     active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
     active(U153(tt())) -> mark(tt())
     active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
     active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
     active(U181(tt(),Y)) -> mark(Y)
     active(U191(tt(),XS)) -> mark(pair(nil(),XS))
     active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
     active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
     active(U21(tt(),X)) -> mark(X)
     active(U211(tt(),XS)) -> mark(XS)
     active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
     active(U31(tt(),N)) -> mark(N)
     active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
     active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
     active(U43(tt())) -> mark(tt())
     active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
     active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
     active(U53(tt())) -> mark(tt())
     active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
     active(U62(tt())) -> mark(tt())
     active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
     active(U72(tt())) -> mark(tt())
     active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
     active(U82(tt())) -> mark(tt())
     active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
     active(U92(tt())) -> mark(tt())
     active(afterNth(N,XS)) ->
     mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
     active(and(tt(),X)) -> mark(X)
     active(fst(pair(X,Y))) ->
     mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
     active(head(cons(N,XS))) ->
     mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
     active(isLNat(nil())) -> mark(tt())
     active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
     active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
     active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
     active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
     active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
     active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
     active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
     active(isLNatKind(nil())) -> mark(tt())
     active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
     active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
     active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
     active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
     active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
     active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
     active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
     active(isNatural(0())) -> mark(tt())
     active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
     active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
     active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
     active(isNaturalKind(0())) -> mark(tt())
     active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
     active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
     active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
     active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
     active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
     active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
     active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
     active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
     active(sel(N,XS)) ->
     mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
     active(snd(pair(X,Y))) ->
     mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
     active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
     active(splitAt(s(N),cons(X,XS))) ->
     mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                          and(isLNat(XS),isLNatKind(XS)))),
               N,X,XS))
     active(tail(cons(N,XS))) ->
     mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
     active(take(N,XS)) ->
     mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
     active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
     active(U102(X1,X2)) -> U102(active(X1),X2)
     active(U103(X)) -> U103(active(X))
     active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
     active(snd(X)) -> snd(active(X))
     active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
     active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
     active(U111(X1,X2)) -> U111(active(X1),X2)
     active(U112(X)) -> U112(active(X))
     active(U121(X1,X2)) -> U121(active(X1),X2)
     active(U122(X)) -> U122(active(X))
     active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
     active(U132(X1,X2)) -> U132(active(X1),X2)
     active(U133(X)) -> U133(active(X))
     active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
     active(U142(X1,X2)) -> U142(active(X1),X2)
     active(U143(X)) -> U143(active(X))
     active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
     active(U152(X1,X2)) -> U152(active(X1),X2)
     active(U153(X)) -> U153(active(X))
     active(U161(X1,X2)) -> U161(active(X1),X2)
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(natsFrom(X)) -> natsFrom(active(X))
     active(s(X)) -> s(active(X))
     active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
     active(head(X)) -> head(active(X))
     active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
     active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
     active(U181(X1,X2)) -> U181(active(X1),X2)
     active(U191(X1,X2)) -> U191(active(X1),X2)
     active(pair(X1,X2)) -> pair(active(X1),X2)
     active(pair(X1,X2)) -> pair(X1,active(X2))
     active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
     active(U202(X1,X2)) -> U202(active(X1),X2)
     active(U21(X1,X2)) -> U21(active(X1),X2)
     active(U211(X1,X2)) -> U211(active(X1),X2)
     active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
     active(fst(X)) -> fst(active(X))
     active(U31(X1,X2)) -> U31(active(X1),X2)
     active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
     active(U42(X1,X2)) -> U42(active(X1),X2)
     active(U43(X)) -> U43(active(X))
     active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
     active(U52(X1,X2)) -> U52(active(X1),X2)
     active(U53(X)) -> U53(active(X))
     active(U61(X1,X2)) -> U61(active(X1),X2)
     active(U62(X)) -> U62(active(X))
     active(U71(X1,X2)) -> U71(active(X1),X2)
     active(U72(X)) -> U72(active(X))
     active(U81(X1,X2)) -> U81(active(X1),X2)
     active(U82(X)) -> U82(active(X))
     active(U91(X1,X2)) -> U91(active(X1),X2)
     active(U92(X)) -> U92(active(X))
     active(and(X1,X2)) -> and(active(X1),X2)
     active(tail(X)) -> tail(active(X))
     active(take(X1,X2)) -> take(active(X1),X2)
     active(take(X1,X2)) -> take(X1,active(X2))
     active(sel(X1,X2)) -> sel(active(X1),X2)
     active(sel(X1,X2)) -> sel(X1,active(X2))
     U102(mark(X1),X2) -> mark(U102(X1,X2))
     U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
     U103(mark(X)) -> mark(U103(X))
     U103(ok(X)) -> ok(U103(X))
     snd(mark(X)) -> mark(snd(X))
     snd(ok(X)) -> ok(snd(X))
     U112(mark(X)) -> mark(U112(X))
     U112(ok(X)) -> ok(U112(X))
     U122(mark(X)) -> mark(U122(X))
     U122(ok(X)) -> ok(U122(X))
     U132(mark(X1),X2) -> mark(U132(X1,X2))
     U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
     U133(mark(X)) -> mark(U133(X))
     U133(ok(X)) -> ok(U133(X))
     U142(mark(X1),X2) -> mark(U142(X1,X2))
     U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
     U143(mark(X)) -> mark(U143(X))
     U143(ok(X)) -> ok(U143(X))
     U152(mark(X1),X2) -> mark(U152(X1,X2))
     U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
     U153(mark(X)) -> mark(U153(X))
     U153(ok(X)) -> ok(U153(X))
     head(mark(X)) -> mark(head(X))
     head(ok(X)) -> ok(head(X))
     pair(X1,mark(X2)) -> mark(pair(X1,X2))
     pair(mark(X1),X2) -> mark(pair(X1,X2))
     pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
     U202(mark(X1),X2) -> mark(U202(X1,X2))
     U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
     fst(mark(X)) -> mark(fst(X))
     fst(ok(X)) -> ok(fst(X))
     U42(mark(X1),X2) -> mark(U42(X1,X2))
     U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
     U43(mark(X)) -> mark(U43(X))
     U43(ok(X)) -> ok(U43(X))
     U52(mark(X1),X2) -> mark(U52(X1,X2))
     U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
     U53(mark(X)) -> mark(U53(X))
     U53(ok(X)) -> ok(U53(X))
     U62(mark(X)) -> mark(U62(X))
     U62(ok(X)) -> ok(U62(X))
     U72(mark(X)) -> mark(U72(X))
     U72(ok(X)) -> ok(U72(X))
     U82(mark(X)) -> mark(U82(X))
     U82(ok(X)) -> ok(U82(X))
     U92(mark(X)) -> mark(U92(X))
     U92(ok(X)) -> ok(U92(X))
     U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
     U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
     U21(mark(X1),X2) -> mark(U21(X1,X2))
     U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
     U31(mark(X1),X2) -> mark(U31(X1,X2))
     U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
     U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
     U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
     U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
     U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
     U61(mark(X1),X2) -> mark(U61(X1,X2))
     U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
     U71(mark(X1),X2) -> mark(U71(X1,X2))
     U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
     U81(mark(X1),X2) -> mark(U81(X1,X2))
     U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
     U91(mark(X1),X2) -> mark(U91(X1,X2))
     U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
     U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
     U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
     U111(mark(X1),X2) -> mark(U111(X1,X2))
     U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
     U121(mark(X1),X2) -> mark(U121(X1,X2))
     U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
     U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
     U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
     U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
     U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
     U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
     U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
     U161(mark(X1),X2) -> mark(U161(X1,X2))
     U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
     U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
     U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
     U181(mark(X1),X2) -> mark(U181(X1,X2))
     U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
     U191(mark(X1),X2) -> mark(U191(X1,X2))
     U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
     U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
     U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
     U211(mark(X1),X2) -> mark(U211(X1,X2))
     U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
     U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
     U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
     tail(mark(X)) -> mark(tail(X))
     tail(ok(X)) -> ok(tail(X))
     take(mark(X1),X2) -> mark(take(X1,X2))
     take(X1,mark(X2)) -> mark(take(X1,X2))
     take(ok(X1),ok(X2)) -> ok(take(X1,X2))
     sel(mark(X1),X2) -> mark(sel(X1,X2))
     sel(X1,mark(X2)) -> mark(sel(X1,X2))
     sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
     proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
     proper(tt()) -> ok(tt())
     proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
     proper(isNatural(X)) -> isNatural(proper(X))
     proper(U103(X)) -> U103(proper(X))
     proper(isLNat(X)) -> isLNat(proper(X))
     proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
     proper(snd(X)) -> snd(proper(X))
     proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
     proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
     proper(U112(X)) -> U112(proper(X))
     proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
     proper(U122(X)) -> U122(proper(X))
     proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
     proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
     proper(U133(X)) -> U133(proper(X))
     proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
     proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
     proper(U143(X)) -> U143(proper(X))
     proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
     proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
     proper(U153(X)) -> U153(proper(X))
     proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(natsFrom(X)) -> natsFrom(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
     proper(head(X)) -> head(proper(X))
     proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
     proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
     proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
     proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
     proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
     proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
     proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
     proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
     proper(fst(X)) -> fst(proper(X))
     proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
     proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
     proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
     proper(U43(X)) -> U43(proper(X))
     proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
     proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
     proper(U53(X)) -> U53(proper(X))
     proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
     proper(U62(X)) -> U62(proper(X))
     proper(isPLNat(X)) -> isPLNat(proper(X))
     proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
     proper(U72(X)) -> U72(proper(X))
     proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
     proper(U82(X)) -> U82(proper(X))
     proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
     proper(U92(X)) -> U92(proper(X))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
     proper(isLNatKind(X)) -> isLNatKind(proper(X))
     proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
     proper(tail(X)) -> tail(proper(X))
     proper(take(X1,X2)) -> take(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
    graph:
     ...
    Restore Modifier:
     DPs:
      active#(U101(tt(),V1,V2)) -> isNatural#(V1)
      active#(U101(tt(),V1,V2)) -> U102#(isNatural(V1),V2)
      active#(U102(tt(),V2)) -> isLNat#(V2)
      active#(U102(tt(),V2)) -> U103#(isLNat(V2))
      active#(U11(tt(),N,XS)) -> splitAt#(N,XS)
      active#(U11(tt(),N,XS)) -> snd#(splitAt(N,XS))
      active#(U111(tt(),V1)) -> isLNat#(V1)
      active#(U111(tt(),V1)) -> U112#(isLNat(V1))
      active#(U121(tt(),V1)) -> isNatural#(V1)
      active#(U121(tt(),V1)) -> U122#(isNatural(V1))
      active#(U131(tt(),V1,V2)) -> isNatural#(V1)
      active#(U131(tt(),V1,V2)) -> U132#(isNatural(V1),V2)
      active#(U132(tt(),V2)) -> isLNat#(V2)
      active#(U132(tt(),V2)) -> U133#(isLNat(V2))
      active#(U141(tt(),V1,V2)) -> isLNat#(V1)
      active#(U141(tt(),V1,V2)) -> U142#(isLNat(V1),V2)
      active#(U142(tt(),V2)) -> isLNat#(V2)
      active#(U142(tt(),V2)) -> U143#(isLNat(V2))
      active#(U151(tt(),V1,V2)) -> isNatural#(V1)
      active#(U151(tt(),V1,V2)) -> U152#(isNatural(V1),V2)
      active#(U152(tt(),V2)) -> isLNat#(V2)
      active#(U152(tt(),V2)) -> U153#(isLNat(V2))
      active#(U161(tt(),N)) -> s#(N)
      active#(U161(tt(),N)) -> natsFrom#(s(N))
      active#(U161(tt(),N)) -> cons#(N,natsFrom(s(N)))
      active#(U171(tt(),N,XS)) -> afterNth#(N,XS)
      active#(U171(tt(),N,XS)) -> head#(afterNth(N,XS))
      active#(U191(tt(),XS)) -> pair#(nil(),XS)
      active#(U201(tt(),N,X,XS)) -> splitAt#(N,XS)
      active#(U201(tt(),N,X,XS)) -> U202#(splitAt(N,XS),X)
      active#(U202(pair(YS,ZS),X)) -> cons#(X,YS)
      active#(U202(pair(YS,ZS),X)) -> pair#(cons(X,YS),ZS)
      active#(U221(tt(),N,XS)) -> splitAt#(N,XS)
      active#(U221(tt(),N,XS)) -> fst#(splitAt(N,XS))
      active#(U41(tt(),V1,V2)) -> isNatural#(V1)
      active#(U41(tt(),V1,V2)) -> U42#(isNatural(V1),V2)
      active#(U42(tt(),V2)) -> isLNat#(V2)
      active#(U42(tt(),V2)) -> U43#(isLNat(V2))
      active#(U51(tt(),V1,V2)) -> isNatural#(V1)
      active#(U51(tt(),V1,V2)) -> U52#(isNatural(V1),V2)
      active#(U52(tt(),V2)) -> isLNat#(V2)
      active#(U52(tt(),V2)) -> U53#(isLNat(V2))
      active#(U61(tt(),V1)) -> isPLNat#(V1)
      active#(U61(tt(),V1)) -> U62#(isPLNat(V1))
      active#(U71(tt(),V1)) -> isNatural#(V1)
      active#(U71(tt(),V1)) -> U72#(isNatural(V1))
      active#(U81(tt(),V1)) -> isPLNat#(V1)
      active#(U81(tt(),V1)) -> U82#(isPLNat(V1))
      active#(U91(tt(),V1)) -> isLNat#(V1)
      active#(U91(tt(),V1)) -> U92#(isLNat(V1))
      active#(afterNth(N,XS)) -> isLNatKind#(XS)
      active#(afterNth(N,XS)) -> isLNat#(XS)
      active#(afterNth(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
      active#(afterNth(N,XS)) -> isNaturalKind#(N)
      active#(afterNth(N,XS)) -> isNatural#(N)
      active#(afterNth(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
      active#(afterNth(N,XS)) ->
      and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
      active#(afterNth(N,XS)) ->
      U11#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
      active#(fst(pair(X,Y))) -> isLNatKind#(Y)
      active#(fst(pair(X,Y))) -> isLNat#(Y)
      active#(fst(pair(X,Y))) -> and#(isLNat(Y),isLNatKind(Y))
      active#(fst(pair(X,Y))) -> isLNatKind#(X)
      active#(fst(pair(X,Y))) -> isLNat#(X)
      active#(fst(pair(X,Y))) -> and#(isLNat(X),isLNatKind(X))
      active#(fst(pair(X,Y))) -> and#(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y)))
      active#(fst(pair(X,Y))) ->
      U21#(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X)
      active#(head(cons(N,XS))) -> isLNatKind#(XS)
      active#(head(cons(N,XS))) -> isLNat#(XS)
      active#(head(cons(N,XS))) -> and#(isLNat(XS),isLNatKind(XS))
      active#(head(cons(N,XS))) -> isNaturalKind#(N)
      active#(head(cons(N,XS))) -> isNatural#(N)
      active#(head(cons(N,XS))) -> and#(isNatural(N),isNaturalKind(N))
      active#(head(cons(N,XS))) ->
      and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
      active#(head(cons(N,XS))) ->
      U31#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N)
      active#(isLNat(afterNth(V1,V2))) -> isLNatKind#(V2)
      active#(isLNat(afterNth(V1,V2))) -> isNaturalKind#(V1)
      active#(isLNat(afterNth(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isLNat(afterNth(V1,V2))) -> U41#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
      active#(isLNat(cons(V1,V2))) -> isLNatKind#(V2)
      active#(isLNat(cons(V1,V2))) -> isNaturalKind#(V1)
      active#(isLNat(cons(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isLNat(cons(V1,V2))) -> U51#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
      active#(isLNat(fst(V1))) -> isPLNatKind#(V1)
      active#(isLNat(fst(V1))) -> U61#(isPLNatKind(V1),V1)
      active#(isLNat(natsFrom(V1))) -> isNaturalKind#(V1)
      active#(isLNat(natsFrom(V1))) -> U71#(isNaturalKind(V1),V1)
      active#(isLNat(snd(V1))) -> isPLNatKind#(V1)
      active#(isLNat(snd(V1))) -> U81#(isPLNatKind(V1),V1)
      active#(isLNat(tail(V1))) -> isLNatKind#(V1)
      active#(isLNat(tail(V1))) -> U91#(isLNatKind(V1),V1)
      active#(isLNat(take(V1,V2))) -> isLNatKind#(V2)
      active#(isLNat(take(V1,V2))) -> isNaturalKind#(V1)
      active#(isLNat(take(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isLNat(take(V1,V2))) -> U101#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
      active#(isLNatKind(afterNth(V1,V2))) -> isLNatKind#(V2)
      active#(isLNatKind(afterNth(V1,V2))) -> isNaturalKind#(V1)
      active#(isLNatKind(afterNth(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isLNatKind(cons(V1,V2))) -> isLNatKind#(V2)
      active#(isLNatKind(cons(V1,V2))) -> isNaturalKind#(V1)
      active#(isLNatKind(cons(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isLNatKind(fst(V1))) -> isPLNatKind#(V1)
      active#(isLNatKind(natsFrom(V1))) -> isNaturalKind#(V1)
      active#(isLNatKind(snd(V1))) -> isPLNatKind#(V1)
      active#(isLNatKind(tail(V1))) -> isLNatKind#(V1)
      active#(isLNatKind(take(V1,V2))) -> isLNatKind#(V2)
      active#(isLNatKind(take(V1,V2))) -> isNaturalKind#(V1)
      active#(isLNatKind(take(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isNatural(head(V1))) -> isLNatKind#(V1)
      active#(isNatural(head(V1))) -> U111#(isLNatKind(V1),V1)
      active#(isNatural(s(V1))) -> isNaturalKind#(V1)
      active#(isNatural(s(V1))) -> U121#(isNaturalKind(V1),V1)
      active#(isNatural(sel(V1,V2))) -> isLNatKind#(V2)
      active#(isNatural(sel(V1,V2))) -> isNaturalKind#(V1)
      active#(isNatural(sel(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isNatural(sel(V1,V2))) -> U131#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
      active#(isNaturalKind(head(V1))) -> isLNatKind#(V1)
      active#(isNaturalKind(s(V1))) -> isNaturalKind#(V1)
      active#(isNaturalKind(sel(V1,V2))) -> isLNatKind#(V2)
      active#(isNaturalKind(sel(V1,V2))) -> isNaturalKind#(V1)
      active#(isNaturalKind(sel(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isPLNat(pair(V1,V2))) -> isLNatKind#(V2)
      active#(isPLNat(pair(V1,V2))) -> isLNatKind#(V1)
      active#(isPLNat(pair(V1,V2))) -> and#(isLNatKind(V1),isLNatKind(V2))
      active#(isPLNat(pair(V1,V2))) -> U141#(and(isLNatKind(V1),isLNatKind(V2)),V1,V2)
      active#(isPLNat(splitAt(V1,V2))) -> isLNatKind#(V2)
      active#(isPLNat(splitAt(V1,V2))) -> isNaturalKind#(V1)
      active#(isPLNat(splitAt(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(isPLNat(splitAt(V1,V2))) -> U151#(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2)
      active#(isPLNatKind(pair(V1,V2))) -> isLNatKind#(V2)
      active#(isPLNatKind(pair(V1,V2))) -> isLNatKind#(V1)
      active#(isPLNatKind(pair(V1,V2))) -> and#(isLNatKind(V1),isLNatKind(V2))
      active#(isPLNatKind(splitAt(V1,V2))) -> isLNatKind#(V2)
      active#(isPLNatKind(splitAt(V1,V2))) -> isNaturalKind#(V1)
      active#(isPLNatKind(splitAt(V1,V2))) -> and#(isNaturalKind(V1),isLNatKind(V2))
      active#(natsFrom(N)) -> isNaturalKind#(N)
      active#(natsFrom(N)) -> isNatural#(N)
      active#(natsFrom(N)) -> and#(isNatural(N),isNaturalKind(N))
      active#(natsFrom(N)) -> U161#(and(isNatural(N),isNaturalKind(N)),N)
      active#(sel(N,XS)) -> isLNatKind#(XS)
      active#(sel(N,XS)) -> isLNat#(XS)
      active#(sel(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
      active#(sel(N,XS)) -> isNaturalKind#(N)
      active#(sel(N,XS)) -> isNatural#(N)
      active#(sel(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
      active#(sel(N,XS)) -> and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
      active#(sel(N,XS)) ->
      U171#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
      active#(snd(pair(X,Y))) -> isLNatKind#(Y)
      active#(snd(pair(X,Y))) -> isLNat#(Y)
      active#(snd(pair(X,Y))) -> and#(isLNat(Y),isLNatKind(Y))
      active#(snd(pair(X,Y))) -> isLNatKind#(X)
      active#(snd(pair(X,Y))) -> isLNat#(X)
      active#(snd(pair(X,Y))) -> and#(isLNat(X),isLNatKind(X))
      active#(snd(pair(X,Y))) -> and#(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y)))
      active#(snd(pair(X,Y))) ->
      U181#(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y)
      active#(splitAt(0(),XS)) -> isLNatKind#(XS)
      active#(splitAt(0(),XS)) -> isLNat#(XS)
      active#(splitAt(0(),XS)) -> and#(isLNat(XS),isLNatKind(XS))
      active#(splitAt(0(),XS)) -> U191#(and(isLNat(XS),isLNatKind(XS)),XS)
      active#(splitAt(s(N),cons(X,XS))) -> isLNatKind#(XS)
      active#(splitAt(s(N),cons(X,XS))) -> isLNat#(XS)
      active#(splitAt(s(N),cons(X,XS))) -> and#(isLNat(XS),isLNatKind(XS))
      active#(splitAt(s(N),cons(X,XS))) -> isNaturalKind#(X)
      active#(splitAt(s(N),cons(X,XS))) -> isNatural#(X)
      active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(X),isNaturalKind(X))
      active#(splitAt(s(N),cons(X,XS))) ->
      and#(and(isNatural(X),isNaturalKind(X)),and(isLNat(XS),isLNatKind(XS)))
      active#(splitAt(s(N),cons(X,XS))) -> isNaturalKind#(N)
      active#(splitAt(s(N),cons(X,XS))) -> isNatural#(N)
      active#(splitAt(s(N),cons(X,XS))) -> and#(isNatural(N),isNaturalKind(N))
      active#(splitAt(s(N),cons(X,XS))) ->
      and#(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                  and(isLNat(XS),isLNatKind(XS))))
      active#(splitAt(s(N),cons(X,XS))) ->
      U201#(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                       and(isLNat(XS),isLNatKind(XS)))),
            N,X,XS)
      active#(tail(cons(N,XS))) -> isLNatKind#(XS)
      active#(tail(cons(N,XS))) -> isLNat#(XS)
      active#(tail(cons(N,XS))) -> and#(isLNat(XS),isLNatKind(XS))
      active#(tail(cons(N,XS))) -> isNaturalKind#(N)
      active#(tail(cons(N,XS))) -> isNatural#(N)
      active#(tail(cons(N,XS))) -> and#(isNatural(N),isNaturalKind(N))
      active#(tail(cons(N,XS))) ->
      and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
      active#(tail(cons(N,XS))) ->
      U211#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS)
      active#(take(N,XS)) -> isLNatKind#(XS)
      active#(take(N,XS)) -> isLNat#(XS)
      active#(take(N,XS)) -> and#(isLNat(XS),isLNatKind(XS))
      active#(take(N,XS)) -> isNaturalKind#(N)
      active#(take(N,XS)) -> isNatural#(N)
      active#(take(N,XS)) -> and#(isNatural(N),isNaturalKind(N))
      active#(take(N,XS)) ->
      and#(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS)))
      active#(take(N,XS)) ->
      U221#(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
      active#(U101(X1,X2,X3)) -> active#(X1)
      active#(U101(X1,X2,X3)) -> U101#(active(X1),X2,X3)
      active#(U102(X1,X2)) -> active#(X1)
      active#(U102(X1,X2)) -> U102#(active(X1),X2)
      active#(U103(X)) -> active#(X)
      active#(U103(X)) -> U103#(active(X))
      active#(U11(X1,X2,X3)) -> active#(X1)
      active#(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
      active#(snd(X)) -> active#(X)
      active#(snd(X)) -> snd#(active(X))
      active#(splitAt(X1,X2)) -> active#(X1)
      active#(splitAt(X1,X2)) -> splitAt#(active(X1),X2)
      active#(splitAt(X1,X2)) -> active#(X2)
      active#(splitAt(X1,X2)) -> splitAt#(X1,active(X2))
      active#(U111(X1,X2)) -> active#(X1)
      active#(U111(X1,X2)) -> U111#(active(X1),X2)
      active#(U112(X)) -> active#(X)
      active#(U112(X)) -> U112#(active(X))
      active#(U121(X1,X2)) -> active#(X1)
      active#(U121(X1,X2)) -> U121#(active(X1),X2)
      active#(U122(X)) -> active#(X)
      active#(U122(X)) -> U122#(active(X))
      active#(U131(X1,X2,X3)) -> active#(X1)
      active#(U131(X1,X2,X3)) -> U131#(active(X1),X2,X3)
      active#(U132(X1,X2)) -> active#(X1)
      active#(U132(X1,X2)) -> U132#(active(X1),X2)
      active#(U133(X)) -> active#(X)
      active#(U133(X)) -> U133#(active(X))
      active#(U141(X1,X2,X3)) -> active#(X1)
      active#(U141(X1,X2,X3)) -> U141#(active(X1),X2,X3)
      active#(U142(X1,X2)) -> active#(X1)
      active#(U142(X1,X2)) -> U142#(active(X1),X2)
      active#(U143(X)) -> active#(X)
      active#(U143(X)) -> U143#(active(X))
      active#(U151(X1,X2,X3)) -> active#(X1)
      active#(U151(X1,X2,X3)) -> U151#(active(X1),X2,X3)
      active#(U152(X1,X2)) -> active#(X1)
      active#(U152(X1,X2)) -> U152#(active(X1),X2)
      active#(U153(X)) -> active#(X)
      active#(U153(X)) -> U153#(active(X))
      active#(U161(X1,X2)) -> active#(X1)
      active#(U161(X1,X2)) -> U161#(active(X1),X2)
      active#(cons(X1,X2)) -> active#(X1)
      active#(cons(X1,X2)) -> cons#(active(X1),X2)
      active#(natsFrom(X)) -> active#(X)
      active#(natsFrom(X)) -> natsFrom#(active(X))
      active#(s(X)) -> active#(X)
      active#(s(X)) -> s#(active(X))
      active#(U171(X1,X2,X3)) -> active#(X1)
      active#(U171(X1,X2,X3)) -> U171#(active(X1),X2,X3)
      active#(head(X)) -> active#(X)
      active#(head(X)) -> head#(active(X))
      active#(afterNth(X1,X2)) -> active#(X1)
      active#(afterNth(X1,X2)) -> afterNth#(active(X1),X2)
      active#(afterNth(X1,X2)) -> active#(X2)
      active#(afterNth(X1,X2)) -> afterNth#(X1,active(X2))
      active#(U181(X1,X2)) -> active#(X1)
      active#(U181(X1,X2)) -> U181#(active(X1),X2)
      active#(U191(X1,X2)) -> active#(X1)
      active#(U191(X1,X2)) -> U191#(active(X1),X2)
      active#(pair(X1,X2)) -> active#(X1)
      active#(pair(X1,X2)) -> pair#(active(X1),X2)
      active#(pair(X1,X2)) -> active#(X2)
      active#(pair(X1,X2)) -> pair#(X1,active(X2))
      active#(U201(X1,X2,X3,X4)) -> active#(X1)
      active#(U201(X1,X2,X3,X4)) -> U201#(active(X1),X2,X3,X4)
      active#(U202(X1,X2)) -> active#(X1)
      active#(U202(X1,X2)) -> U202#(active(X1),X2)
      active#(U21(X1,X2)) -> active#(X1)
      active#(U21(X1,X2)) -> U21#(active(X1),X2)
      active#(U211(X1,X2)) -> active#(X1)
      active#(U211(X1,X2)) -> U211#(active(X1),X2)
      active#(U221(X1,X2,X3)) -> active#(X1)
      active#(U221(X1,X2,X3)) -> U221#(active(X1),X2,X3)
      active#(fst(X)) -> active#(X)
      active#(fst(X)) -> fst#(active(X))
      active#(U31(X1,X2)) -> active#(X1)
      active#(U31(X1,X2)) -> U31#(active(X1),X2)
      active#(U41(X1,X2,X3)) -> active#(X1)
      active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3)
      active#(U42(X1,X2)) -> active#(X1)
      active#(U42(X1,X2)) -> U42#(active(X1),X2)
      active#(U43(X)) -> active#(X)
      active#(U43(X)) -> U43#(active(X))
      active#(U51(X1,X2,X3)) -> active#(X1)
      active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3)
      active#(U52(X1,X2)) -> active#(X1)
      active#(U52(X1,X2)) -> U52#(active(X1),X2)
      active#(U53(X)) -> active#(X)
      active#(U53(X)) -> U53#(active(X))
      active#(U61(X1,X2)) -> active#(X1)
      active#(U61(X1,X2)) -> U61#(active(X1),X2)
      active#(U62(X)) -> active#(X)
      active#(U62(X)) -> U62#(active(X))
      active#(U71(X1,X2)) -> active#(X1)
      active#(U71(X1,X2)) -> U71#(active(X1),X2)
      active#(U72(X)) -> active#(X)
      active#(U72(X)) -> U72#(active(X))
      active#(U81(X1,X2)) -> active#(X1)
      active#(U81(X1,X2)) -> U81#(active(X1),X2)
      active#(U82(X)) -> active#(X)
      active#(U82(X)) -> U82#(active(X))
      active#(U91(X1,X2)) -> active#(X1)
      active#(U91(X1,X2)) -> U91#(active(X1),X2)
      active#(U92(X)) -> active#(X)
      active#(U92(X)) -> U92#(active(X))
      active#(and(X1,X2)) -> active#(X1)
      active#(and(X1,X2)) -> and#(active(X1),X2)
      active#(tail(X)) -> active#(X)
      active#(tail(X)) -> tail#(active(X))
      active#(take(X1,X2)) -> active#(X1)
      active#(take(X1,X2)) -> take#(active(X1),X2)
      active#(take(X1,X2)) -> active#(X2)
      active#(take(X1,X2)) -> take#(X1,active(X2))
      active#(sel(X1,X2)) -> active#(X1)
      active#(sel(X1,X2)) -> sel#(active(X1),X2)
      active#(sel(X1,X2)) -> active#(X2)
      active#(sel(X1,X2)) -> sel#(X1,active(X2))
      U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
      U102#(mark(X1),X2) -> U102#(X1,X2)
      U103#(mark(X)) -> U103#(X)
      U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
      snd#(mark(X)) -> snd#(X)
      splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
      splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
      U111#(mark(X1),X2) -> U111#(X1,X2)
      U112#(mark(X)) -> U112#(X)
      U121#(mark(X1),X2) -> U121#(X1,X2)
      U122#(mark(X)) -> U122#(X)
      U131#(mark(X1),X2,X3) -> U131#(X1,X2,X3)
      U132#(mark(X1),X2) -> U132#(X1,X2)
      U133#(mark(X)) -> U133#(X)
      U141#(mark(X1),X2,X3) -> U141#(X1,X2,X3)
      U142#(mark(X1),X2) -> U142#(X1,X2)
      U143#(mark(X)) -> U143#(X)
      U151#(mark(X1),X2,X3) -> U151#(X1,X2,X3)
      U152#(mark(X1),X2) -> U152#(X1,X2)
      U153#(mark(X)) -> U153#(X)
      U161#(mark(X1),X2) -> U161#(X1,X2)
      cons#(mark(X1),X2) -> cons#(X1,X2)
      natsFrom#(mark(X)) -> natsFrom#(X)
      s#(mark(X)) -> s#(X)
      U171#(mark(X1),X2,X3) -> U171#(X1,X2,X3)
      head#(mark(X)) -> head#(X)
      afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
      afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
      U181#(mark(X1),X2) -> U181#(X1,X2)
      U191#(mark(X1),X2) -> U191#(X1,X2)
      pair#(mark(X1),X2) -> pair#(X1,X2)
      pair#(X1,mark(X2)) -> pair#(X1,X2)
      U201#(mark(X1),X2,X3,X4) -> U201#(X1,X2,X3,X4)
      U202#(mark(X1),X2) -> U202#(X1,X2)
      U21#(mark(X1),X2) -> U21#(X1,X2)
      U211#(mark(X1),X2) -> U211#(X1,X2)
      U221#(mark(X1),X2,X3) -> U221#(X1,X2,X3)
      fst#(mark(X)) -> fst#(X)
      U31#(mark(X1),X2) -> U31#(X1,X2)
      U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
      U42#(mark(X1),X2) -> U42#(X1,X2)
      U43#(mark(X)) -> U43#(X)
      U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
      U52#(mark(X1),X2) -> U52#(X1,X2)
      U53#(mark(X)) -> U53#(X)
      U61#(mark(X1),X2) -> U61#(X1,X2)
      U62#(mark(X)) -> U62#(X)
      U71#(mark(X1),X2) -> U71#(X1,X2)
      U72#(mark(X)) -> U72#(X)
      U81#(mark(X1),X2) -> U81#(X1,X2)
      U82#(mark(X)) -> U82#(X)
      U91#(mark(X1),X2) -> U91#(X1,X2)
      U92#(mark(X)) -> U92#(X)
      and#(mark(X1),X2) -> and#(X1,X2)
      tail#(mark(X)) -> tail#(X)
      take#(mark(X1),X2) -> take#(X1,X2)
      take#(X1,mark(X2)) -> take#(X1,X2)
      sel#(mark(X1),X2) -> sel#(X1,X2)
      sel#(X1,mark(X2)) -> sel#(X1,X2)
      proper#(U101(X1,X2,X3)) -> proper#(X3)
      proper#(U101(X1,X2,X3)) -> proper#(X2)
      proper#(U101(X1,X2,X3)) -> proper#(X1)
      proper#(U101(X1,X2,X3)) -> U101#(proper(X1),proper(X2),proper(X3))
      proper#(U102(X1,X2)) -> proper#(X2)
      proper#(U102(X1,X2)) -> proper#(X1)
      proper#(U102(X1,X2)) -> U102#(proper(X1),proper(X2))
      proper#(isNatural(X)) -> proper#(X)
      proper#(isNatural(X)) -> isNatural#(proper(X))
      proper#(U103(X)) -> proper#(X)
      proper#(U103(X)) -> U103#(proper(X))
      proper#(isLNat(X)) -> proper#(X)
      proper#(isLNat(X)) -> isLNat#(proper(X))
      proper#(U11(X1,X2,X3)) -> proper#(X3)
      proper#(U11(X1,X2,X3)) -> proper#(X2)
      proper#(U11(X1,X2,X3)) -> proper#(X1)
      proper#(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
      proper#(snd(X)) -> proper#(X)
      proper#(snd(X)) -> snd#(proper(X))
      proper#(splitAt(X1,X2)) -> proper#(X2)
      proper#(splitAt(X1,X2)) -> proper#(X1)
      proper#(splitAt(X1,X2)) -> splitAt#(proper(X1),proper(X2))
      proper#(U111(X1,X2)) -> proper#(X2)
      proper#(U111(X1,X2)) -> proper#(X1)
      proper#(U111(X1,X2)) -> U111#(proper(X1),proper(X2))
      proper#(U112(X)) -> proper#(X)
      proper#(U112(X)) -> U112#(proper(X))
      proper#(U121(X1,X2)) -> proper#(X2)
      proper#(U121(X1,X2)) -> proper#(X1)
      proper#(U121(X1,X2)) -> U121#(proper(X1),proper(X2))
      proper#(U122(X)) -> proper#(X)
      proper#(U122(X)) -> U122#(proper(X))
      proper#(U131(X1,X2,X3)) -> proper#(X3)
      proper#(U131(X1,X2,X3)) -> proper#(X2)
      proper#(U131(X1,X2,X3)) -> proper#(X1)
      proper#(U131(X1,X2,X3)) -> U131#(proper(X1),proper(X2),proper(X3))
      proper#(U132(X1,X2)) -> proper#(X2)
      proper#(U132(X1,X2)) -> proper#(X1)
      proper#(U132(X1,X2)) -> U132#(proper(X1),proper(X2))
      proper#(U133(X)) -> proper#(X)
      proper#(U133(X)) -> U133#(proper(X))
      proper#(U141(X1,X2,X3)) -> proper#(X3)
      proper#(U141(X1,X2,X3)) -> proper#(X2)
      proper#(U141(X1,X2,X3)) -> proper#(X1)
      proper#(U141(X1,X2,X3)) -> U141#(proper(X1),proper(X2),proper(X3))
      proper#(U142(X1,X2)) -> proper#(X2)
      proper#(U142(X1,X2)) -> proper#(X1)
      proper#(U142(X1,X2)) -> U142#(proper(X1),proper(X2))
      proper#(U143(X)) -> proper#(X)
      proper#(U143(X)) -> U143#(proper(X))
      proper#(U151(X1,X2,X3)) -> proper#(X3)
      proper#(U151(X1,X2,X3)) -> proper#(X2)
      proper#(U151(X1,X2,X3)) -> proper#(X1)
      proper#(U151(X1,X2,X3)) -> U151#(proper(X1),proper(X2),proper(X3))
      proper#(U152(X1,X2)) -> proper#(X2)
      proper#(U152(X1,X2)) -> proper#(X1)
      proper#(U152(X1,X2)) -> U152#(proper(X1),proper(X2))
      proper#(U153(X)) -> proper#(X)
      proper#(U153(X)) -> U153#(proper(X))
      proper#(U161(X1,X2)) -> proper#(X2)
      proper#(U161(X1,X2)) -> proper#(X1)
      proper#(U161(X1,X2)) -> U161#(proper(X1),proper(X2))
      proper#(cons(X1,X2)) -> proper#(X2)
      proper#(cons(X1,X2)) -> proper#(X1)
      proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
      proper#(natsFrom(X)) -> proper#(X)
      proper#(natsFrom(X)) -> natsFrom#(proper(X))
      proper#(s(X)) -> proper#(X)
      proper#(s(X)) -> s#(proper(X))
      proper#(U171(X1,X2,X3)) -> proper#(X3)
      proper#(U171(X1,X2,X3)) -> proper#(X2)
      proper#(U171(X1,X2,X3)) -> proper#(X1)
      proper#(U171(X1,X2,X3)) -> U171#(proper(X1),proper(X2),proper(X3))
      proper#(head(X)) -> proper#(X)
      proper#(head(X)) -> head#(proper(X))
      proper#(afterNth(X1,X2)) -> proper#(X2)
      proper#(afterNth(X1,X2)) -> proper#(X1)
      proper#(afterNth(X1,X2)) -> afterNth#(proper(X1),proper(X2))
      proper#(U181(X1,X2)) -> proper#(X2)
      proper#(U181(X1,X2)) -> proper#(X1)
      proper#(U181(X1,X2)) -> U181#(proper(X1),proper(X2))
      proper#(U191(X1,X2)) -> proper#(X2)
      proper#(U191(X1,X2)) -> proper#(X1)
      proper#(U191(X1,X2)) -> U191#(proper(X1),proper(X2))
      proper#(pair(X1,X2)) -> proper#(X2)
      proper#(pair(X1,X2)) -> proper#(X1)
      proper#(pair(X1,X2)) -> pair#(proper(X1),proper(X2))
      proper#(U201(X1,X2,X3,X4)) -> proper#(X4)
      proper#(U201(X1,X2,X3,X4)) -> proper#(X3)
      proper#(U201(X1,X2,X3,X4)) -> proper#(X2)
      proper#(U201(X1,X2,X3,X4)) -> proper#(X1)
      proper#(U201(X1,X2,X3,X4)) -> U201#(proper(X1),proper(X2),proper(X3),proper(X4))
      proper#(U202(X1,X2)) -> proper#(X2)
      proper#(U202(X1,X2)) -> proper#(X1)
      proper#(U202(X1,X2)) -> U202#(proper(X1),proper(X2))
      proper#(U21(X1,X2)) -> proper#(X2)
      proper#(U21(X1,X2)) -> proper#(X1)
      proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2))
      proper#(U211(X1,X2)) -> proper#(X2)
      proper#(U211(X1,X2)) -> proper#(X1)
      proper#(U211(X1,X2)) -> U211#(proper(X1),proper(X2))
      proper#(U221(X1,X2,X3)) -> proper#(X3)
      proper#(U221(X1,X2,X3)) -> proper#(X2)
      proper#(U221(X1,X2,X3)) -> proper#(X1)
      proper#(U221(X1,X2,X3)) -> U221#(proper(X1),proper(X2),proper(X3))
      proper#(fst(X)) -> proper#(X)
      proper#(fst(X)) -> fst#(proper(X))
      proper#(U31(X1,X2)) -> proper#(X2)
      proper#(U31(X1,X2)) -> proper#(X1)
      proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2))
      proper#(U41(X1,X2,X3)) -> proper#(X3)
      proper#(U41(X1,X2,X3)) -> proper#(X2)
      proper#(U41(X1,X2,X3)) -> proper#(X1)
      proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3))
      proper#(U42(X1,X2)) -> proper#(X2)
      proper#(U42(X1,X2)) -> proper#(X1)
      proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2))
      proper#(U43(X)) -> proper#(X)
      proper#(U43(X)) -> U43#(proper(X))
      proper#(U51(X1,X2,X3)) -> proper#(X3)
      proper#(U51(X1,X2,X3)) -> proper#(X2)
      proper#(U51(X1,X2,X3)) -> proper#(X1)
      proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3))
      proper#(U52(X1,X2)) -> proper#(X2)
      proper#(U52(X1,X2)) -> proper#(X1)
      proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2))
      proper#(U53(X)) -> proper#(X)
      proper#(U53(X)) -> U53#(proper(X))
      proper#(U61(X1,X2)) -> proper#(X2)
      proper#(U61(X1,X2)) -> proper#(X1)
      proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2))
      proper#(U62(X)) -> proper#(X)
      proper#(U62(X)) -> U62#(proper(X))
      proper#(isPLNat(X)) -> proper#(X)
      proper#(isPLNat(X)) -> isPLNat#(proper(X))
      proper#(U71(X1,X2)) -> proper#(X2)
      proper#(U71(X1,X2)) -> proper#(X1)
      proper#(U71(X1,X2)) -> U71#(proper(X1),proper(X2))
      proper#(U72(X)) -> proper#(X)
      proper#(U72(X)) -> U72#(proper(X))
      proper#(U81(X1,X2)) -> proper#(X2)
      proper#(U81(X1,X2)) -> proper#(X1)
      proper#(U81(X1,X2)) -> U81#(proper(X1),proper(X2))
      proper#(U82(X)) -> proper#(X)
      proper#(U82(X)) -> U82#(proper(X))
      proper#(U91(X1,X2)) -> proper#(X2)
      proper#(U91(X1,X2)) -> proper#(X1)
      proper#(U91(X1,X2)) -> U91#(proper(X1),proper(X2))
      proper#(U92(X)) -> proper#(X)
      proper#(U92(X)) -> U92#(proper(X))
      proper#(and(X1,X2)) -> proper#(X2)
      proper#(and(X1,X2)) -> proper#(X1)
      proper#(and(X1,X2)) -> and#(proper(X1),proper(X2))
      proper#(isNaturalKind(X)) -> proper#(X)
      proper#(isNaturalKind(X)) -> isNaturalKind#(proper(X))
      proper#(isLNatKind(X)) -> proper#(X)
      proper#(isLNatKind(X)) -> isLNatKind#(proper(X))
      proper#(isPLNatKind(X)) -> proper#(X)
      proper#(isPLNatKind(X)) -> isPLNatKind#(proper(X))
      proper#(tail(X)) -> proper#(X)
      proper#(tail(X)) -> tail#(proper(X))
      proper#(take(X1,X2)) -> proper#(X2)
      proper#(take(X1,X2)) -> proper#(X1)
      proper#(take(X1,X2)) -> take#(proper(X1),proper(X2))
      proper#(sel(X1,X2)) -> proper#(X2)
      proper#(sel(X1,X2)) -> proper#(X1)
      proper#(sel(X1,X2)) -> sel#(proper(X1),proper(X2))
      U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3)
      U102#(ok(X1),ok(X2)) -> U102#(X1,X2)
      isNatural#(ok(X)) -> isNatural#(X)
      U103#(ok(X)) -> U103#(X)
      isLNat#(ok(X)) -> isLNat#(X)
      U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
      snd#(ok(X)) -> snd#(X)
      splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
      U111#(ok(X1),ok(X2)) -> U111#(X1,X2)
      U112#(ok(X)) -> U112#(X)
      U121#(ok(X1),ok(X2)) -> U121#(X1,X2)
      U122#(ok(X)) -> U122#(X)
      U131#(ok(X1),ok(X2),ok(X3)) -> U131#(X1,X2,X3)
      U132#(ok(X1),ok(X2)) -> U132#(X1,X2)
      U133#(ok(X)) -> U133#(X)
      U141#(ok(X1),ok(X2),ok(X3)) -> U141#(X1,X2,X3)
      U142#(ok(X1),ok(X2)) -> U142#(X1,X2)
      U143#(ok(X)) -> U143#(X)
      U151#(ok(X1),ok(X2),ok(X3)) -> U151#(X1,X2,X3)
      U152#(ok(X1),ok(X2)) -> U152#(X1,X2)
      U153#(ok(X)) -> U153#(X)
      U161#(ok(X1),ok(X2)) -> U161#(X1,X2)
      cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
      natsFrom#(ok(X)) -> natsFrom#(X)
      s#(ok(X)) -> s#(X)
      U171#(ok(X1),ok(X2),ok(X3)) -> U171#(X1,X2,X3)
      head#(ok(X)) -> head#(X)
      afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
      U181#(ok(X1),ok(X2)) -> U181#(X1,X2)
      U191#(ok(X1),ok(X2)) -> U191#(X1,X2)
      pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
      U201#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U201#(X1,X2,X3,X4)
      U202#(ok(X1),ok(X2)) -> U202#(X1,X2)
      U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
      U211#(ok(X1),ok(X2)) -> U211#(X1,X2)
      U221#(ok(X1),ok(X2),ok(X3)) -> U221#(X1,X2,X3)
      fst#(ok(X)) -> fst#(X)
      U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
      U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
      U42#(ok(X1),ok(X2)) -> U42#(X1,X2)
      U43#(ok(X)) -> U43#(X)
      U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
      U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
      U53#(ok(X)) -> U53#(X)
      U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
      U62#(ok(X)) -> U62#(X)
      isPLNat#(ok(X)) -> isPLNat#(X)
      U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
      U72#(ok(X)) -> U72#(X)
      U81#(ok(X1),ok(X2)) -> U81#(X1,X2)
      U82#(ok(X)) -> U82#(X)
      U91#(ok(X1),ok(X2)) -> U91#(X1,X2)
      U92#(ok(X)) -> U92#(X)
      and#(ok(X1),ok(X2)) -> and#(X1,X2)
      isNaturalKind#(ok(X)) -> isNaturalKind#(X)
      isLNatKind#(ok(X)) -> isLNatKind#(X)
      isPLNatKind#(ok(X)) -> isPLNatKind#(X)
      tail#(ok(X)) -> tail#(X)
      take#(ok(X1),ok(X2)) -> take#(X1,X2)
      sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
      top#(mark(X)) -> proper#(X)
      top#(mark(X)) -> top#(proper(X))
      top#(ok(X)) -> active#(X)
      top#(ok(X)) -> top#(active(X))
     TRS:
      active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
      active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
      active(U103(tt())) -> mark(tt())
      active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
      active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
      active(U112(tt())) -> mark(tt())
      active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
      active(U122(tt())) -> mark(tt())
      active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
      active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
      active(U133(tt())) -> mark(tt())
      active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
      active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
      active(U143(tt())) -> mark(tt())
      active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
      active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
      active(U153(tt())) -> mark(tt())
      active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
      active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
      active(U181(tt(),Y)) -> mark(Y)
      active(U191(tt(),XS)) -> mark(pair(nil(),XS))
      active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
      active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
      active(U21(tt(),X)) -> mark(X)
      active(U211(tt(),XS)) -> mark(XS)
      active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
      active(U31(tt(),N)) -> mark(N)
      active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
      active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
      active(U43(tt())) -> mark(tt())
      active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
      active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
      active(U53(tt())) -> mark(tt())
      active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
      active(U62(tt())) -> mark(tt())
      active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
      active(U72(tt())) -> mark(tt())
      active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
      active(U82(tt())) -> mark(tt())
      active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
      active(U92(tt())) -> mark(tt())
      active(afterNth(N,XS)) ->
      mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
      active(and(tt(),X)) -> mark(X)
      active(fst(pair(X,Y))) ->
      mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
      active(head(cons(N,XS))) ->
      mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
      active(isLNat(nil())) -> mark(tt())
      active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
      active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
      active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
      active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
      active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
      active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
      active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
      active(isLNatKind(nil())) -> mark(tt())
      active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
      active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
      active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
      active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
      active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
      active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
      active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
      active(isNatural(0())) -> mark(tt())
      active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
      active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
      active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
      active(isNaturalKind(0())) -> mark(tt())
      active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
      active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
      active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
      active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
      active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
      active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
      active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
      active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
      active(sel(N,XS)) ->
      mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
      active(snd(pair(X,Y))) ->
      mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
      active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
      active(splitAt(s(N),cons(X,XS))) ->
      mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                           and(isLNat(XS),isLNatKind(XS)))),
                N,X,XS))
      active(tail(cons(N,XS))) ->
      mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
      active(take(N,XS)) ->
      mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
      active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
      active(U102(X1,X2)) -> U102(active(X1),X2)
      active(U103(X)) -> U103(active(X))
      active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
      active(snd(X)) -> snd(active(X))
      active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
      active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
      active(U111(X1,X2)) -> U111(active(X1),X2)
      active(U112(X)) -> U112(active(X))
      active(U121(X1,X2)) -> U121(active(X1),X2)
      active(U122(X)) -> U122(active(X))
      active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
      active(U132(X1,X2)) -> U132(active(X1),X2)
      active(U133(X)) -> U133(active(X))
      active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
      active(U142(X1,X2)) -> U142(active(X1),X2)
      active(U143(X)) -> U143(active(X))
      active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
      active(U152(X1,X2)) -> U152(active(X1),X2)
      active(U153(X)) -> U153(active(X))
      active(U161(X1,X2)) -> U161(active(X1),X2)
      active(cons(X1,X2)) -> cons(active(X1),X2)
      active(natsFrom(X)) -> natsFrom(active(X))
      active(s(X)) -> s(active(X))
      active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
      active(head(X)) -> head(active(X))
      active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
      active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
      active(U181(X1,X2)) -> U181(active(X1),X2)
      active(U191(X1,X2)) -> U191(active(X1),X2)
      active(pair(X1,X2)) -> pair(active(X1),X2)
      active(pair(X1,X2)) -> pair(X1,active(X2))
      active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
      active(U202(X1,X2)) -> U202(active(X1),X2)
      active(U21(X1,X2)) -> U21(active(X1),X2)
      active(U211(X1,X2)) -> U211(active(X1),X2)
      active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
      active(fst(X)) -> fst(active(X))
      active(U31(X1,X2)) -> U31(active(X1),X2)
      active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
      active(U42(X1,X2)) -> U42(active(X1),X2)
      active(U43(X)) -> U43(active(X))
      active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
      active(U52(X1,X2)) -> U52(active(X1),X2)
      active(U53(X)) -> U53(active(X))
      active(U61(X1,X2)) -> U61(active(X1),X2)
      active(U62(X)) -> U62(active(X))
      active(U71(X1,X2)) -> U71(active(X1),X2)
      active(U72(X)) -> U72(active(X))
      active(U81(X1,X2)) -> U81(active(X1),X2)
      active(U82(X)) -> U82(active(X))
      active(U91(X1,X2)) -> U91(active(X1),X2)
      active(U92(X)) -> U92(active(X))
      active(and(X1,X2)) -> and(active(X1),X2)
      active(tail(X)) -> tail(active(X))
      active(take(X1,X2)) -> take(active(X1),X2)
      active(take(X1,X2)) -> take(X1,active(X2))
      active(sel(X1,X2)) -> sel(active(X1),X2)
      active(sel(X1,X2)) -> sel(X1,active(X2))
      U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
      U102(mark(X1),X2) -> mark(U102(X1,X2))
      U103(mark(X)) -> mark(U103(X))
      U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
      snd(mark(X)) -> mark(snd(X))
      splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
      splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
      U111(mark(X1),X2) -> mark(U111(X1,X2))
      U112(mark(X)) -> mark(U112(X))
      U121(mark(X1),X2) -> mark(U121(X1,X2))
      U122(mark(X)) -> mark(U122(X))
      U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
      U132(mark(X1),X2) -> mark(U132(X1,X2))
      U133(mark(X)) -> mark(U133(X))
      U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
      U142(mark(X1),X2) -> mark(U142(X1,X2))
      U143(mark(X)) -> mark(U143(X))
      U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
      U152(mark(X1),X2) -> mark(U152(X1,X2))
      U153(mark(X)) -> mark(U153(X))
      U161(mark(X1),X2) -> mark(U161(X1,X2))
      cons(mark(X1),X2) -> mark(cons(X1,X2))
      natsFrom(mark(X)) -> mark(natsFrom(X))
      s(mark(X)) -> mark(s(X))
      U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
      head(mark(X)) -> mark(head(X))
      afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
      afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
      U181(mark(X1),X2) -> mark(U181(X1,X2))
      U191(mark(X1),X2) -> mark(U191(X1,X2))
      pair(mark(X1),X2) -> mark(pair(X1,X2))
      pair(X1,mark(X2)) -> mark(pair(X1,X2))
      U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
      U202(mark(X1),X2) -> mark(U202(X1,X2))
      U21(mark(X1),X2) -> mark(U21(X1,X2))
      U211(mark(X1),X2) -> mark(U211(X1,X2))
      U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
      fst(mark(X)) -> mark(fst(X))
      U31(mark(X1),X2) -> mark(U31(X1,X2))
      U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
      U42(mark(X1),X2) -> mark(U42(X1,X2))
      U43(mark(X)) -> mark(U43(X))
      U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
      U52(mark(X1),X2) -> mark(U52(X1,X2))
      U53(mark(X)) -> mark(U53(X))
      U61(mark(X1),X2) -> mark(U61(X1,X2))
      U62(mark(X)) -> mark(U62(X))
      U71(mark(X1),X2) -> mark(U71(X1,X2))
      U72(mark(X)) -> mark(U72(X))
      U81(mark(X1),X2) -> mark(U81(X1,X2))
      U82(mark(X)) -> mark(U82(X))
      U91(mark(X1),X2) -> mark(U91(X1,X2))
      U92(mark(X)) -> mark(U92(X))
      and(mark(X1),X2) -> mark(and(X1,X2))
      tail(mark(X)) -> mark(tail(X))
      take(mark(X1),X2) -> mark(take(X1,X2))
      take(X1,mark(X2)) -> mark(take(X1,X2))
      sel(mark(X1),X2) -> mark(sel(X1,X2))
      sel(X1,mark(X2)) -> mark(sel(X1,X2))
      proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
      proper(tt()) -> ok(tt())
      proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
      proper(isNatural(X)) -> isNatural(proper(X))
      proper(U103(X)) -> U103(proper(X))
      proper(isLNat(X)) -> isLNat(proper(X))
      proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
      proper(snd(X)) -> snd(proper(X))
      proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
      proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
      proper(U112(X)) -> U112(proper(X))
      proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
      proper(U122(X)) -> U122(proper(X))
      proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
      proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
      proper(U133(X)) -> U133(proper(X))
      proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
      proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
      proper(U143(X)) -> U143(proper(X))
      proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
      proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
      proper(U153(X)) -> U153(proper(X))
      proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(natsFrom(X)) -> natsFrom(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
      proper(head(X)) -> head(proper(X))
      proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
      proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
      proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
      proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
      proper(nil()) -> ok(nil())
      proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
      proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
      proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
      proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
      proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
      proper(fst(X)) -> fst(proper(X))
      proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
      proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
      proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
      proper(U43(X)) -> U43(proper(X))
      proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
      proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
      proper(U53(X)) -> U53(proper(X))
      proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
      proper(U62(X)) -> U62(proper(X))
      proper(isPLNat(X)) -> isPLNat(proper(X))
      proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
      proper(U72(X)) -> U72(proper(X))
      proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
      proper(U82(X)) -> U82(proper(X))
      proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
      proper(U92(X)) -> U92(proper(X))
      proper(and(X1,X2)) -> and(proper(X1),proper(X2))
      proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
      proper(isLNatKind(X)) -> isLNatKind(proper(X))
      proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
      proper(tail(X)) -> tail(proper(X))
      proper(take(X1,X2)) -> take(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
      U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
      U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
      isNatural(ok(X)) -> ok(isNatural(X))
      U103(ok(X)) -> ok(U103(X))
      isLNat(ok(X)) -> ok(isLNat(X))
      U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
      snd(ok(X)) -> ok(snd(X))
      splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
      U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
      U112(ok(X)) -> ok(U112(X))
      U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
      U122(ok(X)) -> ok(U122(X))
      U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
      U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
      U133(ok(X)) -> ok(U133(X))
      U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
      U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
      U143(ok(X)) -> ok(U143(X))
      U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
      U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
      U153(ok(X)) -> ok(U153(X))
      U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      natsFrom(ok(X)) -> ok(natsFrom(X))
      s(ok(X)) -> ok(s(X))
      U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
      head(ok(X)) -> ok(head(X))
      afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
      U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
      U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
      pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
      U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
      U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
      U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
      U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
      U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
      fst(ok(X)) -> ok(fst(X))
      U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
      U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
      U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
      U43(ok(X)) -> ok(U43(X))
      U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
      U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
      U53(ok(X)) -> ok(U53(X))
      U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
      U62(ok(X)) -> ok(U62(X))
      isPLNat(ok(X)) -> ok(isPLNat(X))
      U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
      U72(ok(X)) -> ok(U72(X))
      U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
      U82(ok(X)) -> ok(U82(X))
      U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
      U92(ok(X)) -> ok(U92(X))
      and(ok(X1),ok(X2)) -> ok(and(X1,X2))
      isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
      isLNatKind(ok(X)) -> ok(isLNatKind(X))
      isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
      tail(ok(X)) -> ok(tail(X))
      take(ok(X1),ok(X2)) -> ok(take(X1,X2))
      sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     SCC Processor:
      #sccs: 63
      #rules: 287
      #arcs: 37064/352836
      DPs:
       top#(ok(X)) -> top#(active(X))
       top#(mark(X)) -> top#(proper(X))
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       active#(U101(X1,X2,X3)) -> active#(X1)
       active#(U102(X1,X2)) -> active#(X1)
       active#(U103(X)) -> active#(X)
       active#(U11(X1,X2,X3)) -> active#(X1)
       active#(snd(X)) -> active#(X)
       active#(splitAt(X1,X2)) -> active#(X1)
       active#(splitAt(X1,X2)) -> active#(X2)
       active#(U111(X1,X2)) -> active#(X1)
       active#(U112(X)) -> active#(X)
       active#(U121(X1,X2)) -> active#(X1)
       active#(U122(X)) -> active#(X)
       active#(U131(X1,X2,X3)) -> active#(X1)
       active#(U132(X1,X2)) -> active#(X1)
       active#(U133(X)) -> active#(X)
       active#(U141(X1,X2,X3)) -> active#(X1)
       active#(U142(X1,X2)) -> active#(X1)
       active#(U143(X)) -> active#(X)
       active#(U151(X1,X2,X3)) -> active#(X1)
       active#(U152(X1,X2)) -> active#(X1)
       active#(U153(X)) -> active#(X)
       active#(U161(X1,X2)) -> active#(X1)
       active#(cons(X1,X2)) -> active#(X1)
       active#(natsFrom(X)) -> active#(X)
       active#(s(X)) -> active#(X)
       active#(U171(X1,X2,X3)) -> active#(X1)
       active#(head(X)) -> active#(X)
       active#(afterNth(X1,X2)) -> active#(X1)
       active#(afterNth(X1,X2)) -> active#(X2)
       active#(U181(X1,X2)) -> active#(X1)
       active#(U191(X1,X2)) -> active#(X1)
       active#(pair(X1,X2)) -> active#(X1)
       active#(pair(X1,X2)) -> active#(X2)
       active#(U201(X1,X2,X3,X4)) -> active#(X1)
       active#(U202(X1,X2)) -> active#(X1)
       active#(U21(X1,X2)) -> active#(X1)
       active#(U211(X1,X2)) -> active#(X1)
       active#(U221(X1,X2,X3)) -> active#(X1)
       active#(fst(X)) -> active#(X)
       active#(U31(X1,X2)) -> active#(X1)
       active#(U41(X1,X2,X3)) -> active#(X1)
       active#(U42(X1,X2)) -> active#(X1)
       active#(U43(X)) -> active#(X)
       active#(U51(X1,X2,X3)) -> active#(X1)
       active#(U52(X1,X2)) -> active#(X1)
       active#(U53(X)) -> active#(X)
       active#(U61(X1,X2)) -> active#(X1)
       active#(U62(X)) -> active#(X)
       active#(U71(X1,X2)) -> active#(X1)
       active#(U72(X)) -> active#(X)
       active#(U81(X1,X2)) -> active#(X1)
       active#(U82(X)) -> active#(X)
       active#(U91(X1,X2)) -> active#(X1)
       active#(U92(X)) -> active#(X)
       active#(and(X1,X2)) -> active#(X1)
       active#(tail(X)) -> active#(X)
       active#(take(X1,X2)) -> active#(X1)
       active#(take(X1,X2)) -> active#(X2)
       active#(sel(X1,X2)) -> active#(X1)
       active#(sel(X1,X2)) -> active#(X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       proper#(U101(X1,X2,X3)) -> proper#(X3)
       proper#(U101(X1,X2,X3)) -> proper#(X2)
       proper#(U101(X1,X2,X3)) -> proper#(X1)
       proper#(U102(X1,X2)) -> proper#(X2)
       proper#(U102(X1,X2)) -> proper#(X1)
       proper#(isNatural(X)) -> proper#(X)
       proper#(U103(X)) -> proper#(X)
       proper#(isLNat(X)) -> proper#(X)
       proper#(U11(X1,X2,X3)) -> proper#(X3)
       proper#(U11(X1,X2,X3)) -> proper#(X2)
       proper#(U11(X1,X2,X3)) -> proper#(X1)
       proper#(snd(X)) -> proper#(X)
       proper#(splitAt(X1,X2)) -> proper#(X2)
       proper#(splitAt(X1,X2)) -> proper#(X1)
       proper#(U111(X1,X2)) -> proper#(X2)
       proper#(U111(X1,X2)) -> proper#(X1)
       proper#(U112(X)) -> proper#(X)
       proper#(U121(X1,X2)) -> proper#(X2)
       proper#(U121(X1,X2)) -> proper#(X1)
       proper#(U122(X)) -> proper#(X)
       proper#(U131(X1,X2,X3)) -> proper#(X3)
       proper#(U131(X1,X2,X3)) -> proper#(X2)
       proper#(U131(X1,X2,X3)) -> proper#(X1)
       proper#(U132(X1,X2)) -> proper#(X2)
       proper#(U132(X1,X2)) -> proper#(X1)
       proper#(U133(X)) -> proper#(X)
       proper#(U141(X1,X2,X3)) -> proper#(X3)
       proper#(U141(X1,X2,X3)) -> proper#(X2)
       proper#(U141(X1,X2,X3)) -> proper#(X1)
       proper#(U142(X1,X2)) -> proper#(X2)
       proper#(U142(X1,X2)) -> proper#(X1)
       proper#(U143(X)) -> proper#(X)
       proper#(U151(X1,X2,X3)) -> proper#(X3)
       proper#(U151(X1,X2,X3)) -> proper#(X2)
       proper#(U151(X1,X2,X3)) -> proper#(X1)
       proper#(U152(X1,X2)) -> proper#(X2)
       proper#(U152(X1,X2)) -> proper#(X1)
       proper#(U153(X)) -> proper#(X)
       proper#(U161(X1,X2)) -> proper#(X2)
       proper#(U161(X1,X2)) -> proper#(X1)
       proper#(cons(X1,X2)) -> proper#(X2)
       proper#(cons(X1,X2)) -> proper#(X1)
       proper#(natsFrom(X)) -> proper#(X)
       proper#(s(X)) -> proper#(X)
       proper#(U171(X1,X2,X3)) -> proper#(X3)
       proper#(U171(X1,X2,X3)) -> proper#(X2)
       proper#(U171(X1,X2,X3)) -> proper#(X1)
       proper#(head(X)) -> proper#(X)
       proper#(afterNth(X1,X2)) -> proper#(X2)
       proper#(afterNth(X1,X2)) -> proper#(X1)
       proper#(U181(X1,X2)) -> proper#(X2)
       proper#(U181(X1,X2)) -> proper#(X1)
       proper#(U191(X1,X2)) -> proper#(X2)
       proper#(U191(X1,X2)) -> proper#(X1)
       proper#(pair(X1,X2)) -> proper#(X2)
       proper#(pair(X1,X2)) -> proper#(X1)
       proper#(U201(X1,X2,X3,X4)) -> proper#(X4)
       proper#(U201(X1,X2,X3,X4)) -> proper#(X3)
       proper#(U201(X1,X2,X3,X4)) -> proper#(X2)
       proper#(U201(X1,X2,X3,X4)) -> proper#(X1)
       proper#(U202(X1,X2)) -> proper#(X2)
       proper#(U202(X1,X2)) -> proper#(X1)
       proper#(U21(X1,X2)) -> proper#(X2)
       proper#(U21(X1,X2)) -> proper#(X1)
       proper#(U211(X1,X2)) -> proper#(X2)
       proper#(U211(X1,X2)) -> proper#(X1)
       proper#(U221(X1,X2,X3)) -> proper#(X3)
       proper#(U221(X1,X2,X3)) -> proper#(X2)
       proper#(U221(X1,X2,X3)) -> proper#(X1)
       proper#(fst(X)) -> proper#(X)
       proper#(U31(X1,X2)) -> proper#(X2)
       proper#(U31(X1,X2)) -> proper#(X1)
       proper#(U41(X1,X2,X3)) -> proper#(X3)
       proper#(U41(X1,X2,X3)) -> proper#(X2)
       proper#(U41(X1,X2,X3)) -> proper#(X1)
       proper#(U42(X1,X2)) -> proper#(X2)
       proper#(U42(X1,X2)) -> proper#(X1)
       proper#(U43(X)) -> proper#(X)
       proper#(U51(X1,X2,X3)) -> proper#(X3)
       proper#(U51(X1,X2,X3)) -> proper#(X2)
       proper#(U51(X1,X2,X3)) -> proper#(X1)
       proper#(U52(X1,X2)) -> proper#(X2)
       proper#(U52(X1,X2)) -> proper#(X1)
       proper#(U53(X)) -> proper#(X)
       proper#(U61(X1,X2)) -> proper#(X2)
       proper#(U61(X1,X2)) -> proper#(X1)
       proper#(U62(X)) -> proper#(X)
       proper#(isPLNat(X)) -> proper#(X)
       proper#(U71(X1,X2)) -> proper#(X2)
       proper#(U71(X1,X2)) -> proper#(X1)
       proper#(U72(X)) -> proper#(X)
       proper#(U81(X1,X2)) -> proper#(X2)
       proper#(U81(X1,X2)) -> proper#(X1)
       proper#(U82(X)) -> proper#(X)
       proper#(U91(X1,X2)) -> proper#(X2)
       proper#(U91(X1,X2)) -> proper#(X1)
       proper#(U92(X)) -> proper#(X)
       proper#(and(X1,X2)) -> proper#(X2)
       proper#(and(X1,X2)) -> proper#(X1)
       proper#(isNaturalKind(X)) -> proper#(X)
       proper#(isLNatKind(X)) -> proper#(X)
       proper#(isPLNatKind(X)) -> proper#(X)
       proper#(tail(X)) -> proper#(X)
       proper#(take(X1,X2)) -> proper#(X2)
       proper#(take(X1,X2)) -> proper#(X1)
       proper#(sel(X1,X2)) -> proper#(X2)
       proper#(sel(X1,X2)) -> proper#(X1)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       sel#(mark(X1),X2) -> sel#(X1,X2)
       sel#(X1,mark(X2)) -> sel#(X1,X2)
       sel#(ok(X1),ok(X2)) -> sel#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       take#(mark(X1),X2) -> take#(X1,X2)
       take#(X1,mark(X2)) -> take#(X1,X2)
       take#(ok(X1),ok(X2)) -> take#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       tail#(mark(X)) -> tail#(X)
       tail#(ok(X)) -> tail#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isPLNatKind#(ok(X)) -> isPLNatKind#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isLNatKind#(ok(X)) -> isLNatKind#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isNaturalKind#(ok(X)) -> isNaturalKind#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       and#(mark(X1),X2) -> and#(X1,X2)
       and#(ok(X1),ok(X2)) -> and#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U92#(mark(X)) -> U92#(X)
       U92#(ok(X)) -> U92#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U91#(mark(X1),X2) -> U91#(X1,X2)
       U91#(ok(X1),ok(X2)) -> U91#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U82#(mark(X)) -> U82#(X)
       U82#(ok(X)) -> U82#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U81#(mark(X1),X2) -> U81#(X1,X2)
       U81#(ok(X1),ok(X2)) -> U81#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U72#(mark(X)) -> U72#(X)
       U72#(ok(X)) -> U72#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U71#(mark(X1),X2) -> U71#(X1,X2)
       U71#(ok(X1),ok(X2)) -> U71#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isPLNat#(ok(X)) -> isPLNat#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U62#(mark(X)) -> U62#(X)
       U62#(ok(X)) -> U62#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U61#(mark(X1),X2) -> U61#(X1,X2)
       U61#(ok(X1),ok(X2)) -> U61#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U53#(mark(X)) -> U53#(X)
       U53#(ok(X)) -> U53#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U52#(mark(X1),X2) -> U52#(X1,X2)
       U52#(ok(X1),ok(X2)) -> U52#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3)
       U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U43#(mark(X)) -> U43#(X)
       U43#(ok(X)) -> U43#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U42#(mark(X1),X2) -> U42#(X1,X2)
       U42#(ok(X1),ok(X2)) -> U42#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
       U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U31#(mark(X1),X2) -> U31#(X1,X2)
       U31#(ok(X1),ok(X2)) -> U31#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       fst#(mark(X)) -> fst#(X)
       fst#(ok(X)) -> fst#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U221#(mark(X1),X2,X3) -> U221#(X1,X2,X3)
       U221#(ok(X1),ok(X2),ok(X3)) -> U221#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U211#(mark(X1),X2) -> U211#(X1,X2)
       U211#(ok(X1),ok(X2)) -> U211#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U21#(mark(X1),X2) -> U21#(X1,X2)
       U21#(ok(X1),ok(X2)) -> U21#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U202#(mark(X1),X2) -> U202#(X1,X2)
       U202#(ok(X1),ok(X2)) -> U202#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U201#(mark(X1),X2,X3,X4) -> U201#(X1,X2,X3,X4)
       U201#(ok(X1),ok(X2),ok(X3),ok(X4)) -> U201#(X1,X2,X3,X4)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       pair#(mark(X1),X2) -> pair#(X1,X2)
       pair#(X1,mark(X2)) -> pair#(X1,X2)
       pair#(ok(X1),ok(X2)) -> pair#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U191#(mark(X1),X2) -> U191#(X1,X2)
       U191#(ok(X1),ok(X2)) -> U191#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U181#(mark(X1),X2) -> U181#(X1,X2)
       U181#(ok(X1),ok(X2)) -> U181#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       afterNth#(mark(X1),X2) -> afterNth#(X1,X2)
       afterNth#(X1,mark(X2)) -> afterNth#(X1,X2)
       afterNth#(ok(X1),ok(X2)) -> afterNth#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       head#(mark(X)) -> head#(X)
       head#(ok(X)) -> head#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U171#(mark(X1),X2,X3) -> U171#(X1,X2,X3)
       U171#(ok(X1),ok(X2),ok(X3)) -> U171#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       s#(mark(X)) -> s#(X)
       s#(ok(X)) -> s#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       natsFrom#(mark(X)) -> natsFrom#(X)
       natsFrom#(ok(X)) -> natsFrom#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       cons#(mark(X1),X2) -> cons#(X1,X2)
       cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U161#(mark(X1),X2) -> U161#(X1,X2)
       U161#(ok(X1),ok(X2)) -> U161#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U153#(mark(X)) -> U153#(X)
       U153#(ok(X)) -> U153#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U152#(mark(X1),X2) -> U152#(X1,X2)
       U152#(ok(X1),ok(X2)) -> U152#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U151#(mark(X1),X2,X3) -> U151#(X1,X2,X3)
       U151#(ok(X1),ok(X2),ok(X3)) -> U151#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U143#(mark(X)) -> U143#(X)
       U143#(ok(X)) -> U143#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U142#(mark(X1),X2) -> U142#(X1,X2)
       U142#(ok(X1),ok(X2)) -> U142#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U141#(mark(X1),X2,X3) -> U141#(X1,X2,X3)
       U141#(ok(X1),ok(X2),ok(X3)) -> U141#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U133#(mark(X)) -> U133#(X)
       U133#(ok(X)) -> U133#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U132#(mark(X1),X2) -> U132#(X1,X2)
       U132#(ok(X1),ok(X2)) -> U132#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U131#(mark(X1),X2,X3) -> U131#(X1,X2,X3)
       U131#(ok(X1),ok(X2),ok(X3)) -> U131#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U122#(mark(X)) -> U122#(X)
       U122#(ok(X)) -> U122#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U121#(mark(X1),X2) -> U121#(X1,X2)
       U121#(ok(X1),ok(X2)) -> U121#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U112#(mark(X)) -> U112#(X)
       U112#(ok(X)) -> U112#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U111#(mark(X1),X2) -> U111#(X1,X2)
       U111#(ok(X1),ok(X2)) -> U111#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       splitAt#(mark(X1),X2) -> splitAt#(X1,X2)
       splitAt#(X1,mark(X2)) -> splitAt#(X1,X2)
       splitAt#(ok(X1),ok(X2)) -> splitAt#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       snd#(mark(X)) -> snd#(X)
       snd#(ok(X)) -> snd#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
       U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isLNat#(ok(X)) -> isLNat#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U103#(mark(X)) -> U103#(X)
       U103#(ok(X)) -> U103#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       isNatural#(ok(X)) -> isNatural#(X)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U102#(mark(X1),X2) -> U102#(X1,X2)
       U102#(ok(X1),ok(X2)) -> U102#(X1,X2)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
       U101#(ok(X1),ok(X2),ok(X3)) -> U101#(X1,X2,X3)
      TRS:
       active(U101(tt(),V1,V2)) -> mark(U102(isNatural(V1),V2))
       active(U102(tt(),V2)) -> mark(U103(isLNat(V2)))
       active(U103(tt())) -> mark(tt())
       active(U11(tt(),N,XS)) -> mark(snd(splitAt(N,XS)))
       active(U111(tt(),V1)) -> mark(U112(isLNat(V1)))
       active(U112(tt())) -> mark(tt())
       active(U121(tt(),V1)) -> mark(U122(isNatural(V1)))
       active(U122(tt())) -> mark(tt())
       active(U131(tt(),V1,V2)) -> mark(U132(isNatural(V1),V2))
       active(U132(tt(),V2)) -> mark(U133(isLNat(V2)))
       active(U133(tt())) -> mark(tt())
       active(U141(tt(),V1,V2)) -> mark(U142(isLNat(V1),V2))
       active(U142(tt(),V2)) -> mark(U143(isLNat(V2)))
       active(U143(tt())) -> mark(tt())
       active(U151(tt(),V1,V2)) -> mark(U152(isNatural(V1),V2))
       active(U152(tt(),V2)) -> mark(U153(isLNat(V2)))
       active(U153(tt())) -> mark(tt())
       active(U161(tt(),N)) -> mark(cons(N,natsFrom(s(N))))
       active(U171(tt(),N,XS)) -> mark(head(afterNth(N,XS)))
       active(U181(tt(),Y)) -> mark(Y)
       active(U191(tt(),XS)) -> mark(pair(nil(),XS))
       active(U201(tt(),N,X,XS)) -> mark(U202(splitAt(N,XS),X))
       active(U202(pair(YS,ZS),X)) -> mark(pair(cons(X,YS),ZS))
       active(U21(tt(),X)) -> mark(X)
       active(U211(tt(),XS)) -> mark(XS)
       active(U221(tt(),N,XS)) -> mark(fst(splitAt(N,XS)))
       active(U31(tt(),N)) -> mark(N)
       active(U41(tt(),V1,V2)) -> mark(U42(isNatural(V1),V2))
       active(U42(tt(),V2)) -> mark(U43(isLNat(V2)))
       active(U43(tt())) -> mark(tt())
       active(U51(tt(),V1,V2)) -> mark(U52(isNatural(V1),V2))
       active(U52(tt(),V2)) -> mark(U53(isLNat(V2)))
       active(U53(tt())) -> mark(tt())
       active(U61(tt(),V1)) -> mark(U62(isPLNat(V1)))
       active(U62(tt())) -> mark(tt())
       active(U71(tt(),V1)) -> mark(U72(isNatural(V1)))
       active(U72(tt())) -> mark(tt())
       active(U81(tt(),V1)) -> mark(U82(isPLNat(V1)))
       active(U82(tt())) -> mark(tt())
       active(U91(tt(),V1)) -> mark(U92(isLNat(V1)))
       active(U92(tt())) -> mark(tt())
       active(afterNth(N,XS)) ->
       mark(U11(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(and(tt(),X)) -> mark(X)
       active(fst(pair(X,Y))) ->
       mark(U21(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X))
       active(head(cons(N,XS))) ->
       mark(U31(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N))
       active(isLNat(nil())) -> mark(tt())
       active(isLNat(afterNth(V1,V2))) -> mark(U41(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(cons(V1,V2))) -> mark(U51(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1),V1))
       active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1),V1))
       active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1),V1))
       active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1),V1))
       active(isLNat(take(V1,V2))) -> mark(U101(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isLNatKind(nil())) -> mark(tt())
       active(isLNatKind(afterNth(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(cons(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isLNatKind(fst(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(natsFrom(V1))) -> mark(isNaturalKind(V1))
       active(isLNatKind(snd(V1))) -> mark(isPLNatKind(V1))
       active(isLNatKind(tail(V1))) -> mark(isLNatKind(V1))
       active(isLNatKind(take(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isNatural(0())) -> mark(tt())
       active(isNatural(head(V1))) -> mark(U111(isLNatKind(V1),V1))
       active(isNatural(s(V1))) -> mark(U121(isNaturalKind(V1),V1))
       active(isNatural(sel(V1,V2))) -> mark(U131(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isNaturalKind(0())) -> mark(tt())
       active(isNaturalKind(head(V1))) -> mark(isLNatKind(V1))
       active(isNaturalKind(s(V1))) -> mark(isNaturalKind(V1))
       active(isNaturalKind(sel(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(isPLNat(pair(V1,V2))) -> mark(U141(and(isLNatKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNat(splitAt(V1,V2))) -> mark(U151(and(isNaturalKind(V1),isLNatKind(V2)),V1,V2))
       active(isPLNatKind(pair(V1,V2))) -> mark(and(isLNatKind(V1),isLNatKind(V2)))
       active(isPLNatKind(splitAt(V1,V2))) -> mark(and(isNaturalKind(V1),isLNatKind(V2)))
       active(natsFrom(N)) -> mark(U161(and(isNatural(N),isNaturalKind(N)),N))
       active(sel(N,XS)) ->
       mark(U171(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(snd(pair(X,Y))) ->
       mark(U181(and(and(isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y))
       active(splitAt(0(),XS)) -> mark(U191(and(isLNat(XS),isLNatKind(XS)),XS))
       active(splitAt(s(N),cons(X,XS))) ->
       mark(U201(and(and(isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),
                                                            and(isLNat(XS),isLNatKind(XS)))),
                 N,X,XS))
       active(tail(cons(N,XS))) ->
       mark(U211(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS))
       active(take(N,XS)) ->
       mark(U221(and(and(isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS))
       active(U101(X1,X2,X3)) -> U101(active(X1),X2,X3)
       active(U102(X1,X2)) -> U102(active(X1),X2)
       active(U103(X)) -> U103(active(X))
       active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
       active(snd(X)) -> snd(active(X))
       active(splitAt(X1,X2)) -> splitAt(active(X1),X2)
       active(splitAt(X1,X2)) -> splitAt(X1,active(X2))
       active(U111(X1,X2)) -> U111(active(X1),X2)
       active(U112(X)) -> U112(active(X))
       active(U121(X1,X2)) -> U121(active(X1),X2)
       active(U122(X)) -> U122(active(X))
       active(U131(X1,X2,X3)) -> U131(active(X1),X2,X3)
       active(U132(X1,X2)) -> U132(active(X1),X2)
       active(U133(X)) -> U133(active(X))
       active(U141(X1,X2,X3)) -> U141(active(X1),X2,X3)
       active(U142(X1,X2)) -> U142(active(X1),X2)
       active(U143(X)) -> U143(active(X))
       active(U151(X1,X2,X3)) -> U151(active(X1),X2,X3)
       active(U152(X1,X2)) -> U152(active(X1),X2)
       active(U153(X)) -> U153(active(X))
       active(U161(X1,X2)) -> U161(active(X1),X2)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(natsFrom(X)) -> natsFrom(active(X))
       active(s(X)) -> s(active(X))
       active(U171(X1,X2,X3)) -> U171(active(X1),X2,X3)
       active(head(X)) -> head(active(X))
       active(afterNth(X1,X2)) -> afterNth(active(X1),X2)
       active(afterNth(X1,X2)) -> afterNth(X1,active(X2))
       active(U181(X1,X2)) -> U181(active(X1),X2)
       active(U191(X1,X2)) -> U191(active(X1),X2)
       active(pair(X1,X2)) -> pair(active(X1),X2)
       active(pair(X1,X2)) -> pair(X1,active(X2))
       active(U201(X1,X2,X3,X4)) -> U201(active(X1),X2,X3,X4)
       active(U202(X1,X2)) -> U202(active(X1),X2)
       active(U21(X1,X2)) -> U21(active(X1),X2)
       active(U211(X1,X2)) -> U211(active(X1),X2)
       active(U221(X1,X2,X3)) -> U221(active(X1),X2,X3)
       active(fst(X)) -> fst(active(X))
       active(U31(X1,X2)) -> U31(active(X1),X2)
       active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
       active(U42(X1,X2)) -> U42(active(X1),X2)
       active(U43(X)) -> U43(active(X))
       active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3)
       active(U52(X1,X2)) -> U52(active(X1),X2)
       active(U53(X)) -> U53(active(X))
       active(U61(X1,X2)) -> U61(active(X1),X2)
       active(U62(X)) -> U62(active(X))
       active(U71(X1,X2)) -> U71(active(X1),X2)
       active(U72(X)) -> U72(active(X))
       active(U81(X1,X2)) -> U81(active(X1),X2)
       active(U82(X)) -> U82(active(X))
       active(U91(X1,X2)) -> U91(active(X1),X2)
       active(U92(X)) -> U92(active(X))
       active(and(X1,X2)) -> and(active(X1),X2)
       active(tail(X)) -> tail(active(X))
       active(take(X1,X2)) -> take(active(X1),X2)
       active(take(X1,X2)) -> take(X1,active(X2))
       active(sel(X1,X2)) -> sel(active(X1),X2)
       active(sel(X1,X2)) -> sel(X1,active(X2))
       U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
       U102(mark(X1),X2) -> mark(U102(X1,X2))
       U103(mark(X)) -> mark(U103(X))
       U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
       snd(mark(X)) -> mark(snd(X))
       splitAt(mark(X1),X2) -> mark(splitAt(X1,X2))
       splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2))
       U111(mark(X1),X2) -> mark(U111(X1,X2))
       U112(mark(X)) -> mark(U112(X))
       U121(mark(X1),X2) -> mark(U121(X1,X2))
       U122(mark(X)) -> mark(U122(X))
       U131(mark(X1),X2,X3) -> mark(U131(X1,X2,X3))
       U132(mark(X1),X2) -> mark(U132(X1,X2))
       U133(mark(X)) -> mark(U133(X))
       U141(mark(X1),X2,X3) -> mark(U141(X1,X2,X3))
       U142(mark(X1),X2) -> mark(U142(X1,X2))
       U143(mark(X)) -> mark(U143(X))
       U151(mark(X1),X2,X3) -> mark(U151(X1,X2,X3))
       U152(mark(X1),X2) -> mark(U152(X1,X2))
       U153(mark(X)) -> mark(U153(X))
       U161(mark(X1),X2) -> mark(U161(X1,X2))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       natsFrom(mark(X)) -> mark(natsFrom(X))
       s(mark(X)) -> mark(s(X))
       U171(mark(X1),X2,X3) -> mark(U171(X1,X2,X3))
       head(mark(X)) -> mark(head(X))
       afterNth(mark(X1),X2) -> mark(afterNth(X1,X2))
       afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2))
       U181(mark(X1),X2) -> mark(U181(X1,X2))
       U191(mark(X1),X2) -> mark(U191(X1,X2))
       pair(mark(X1),X2) -> mark(pair(X1,X2))
       pair(X1,mark(X2)) -> mark(pair(X1,X2))
       U201(mark(X1),X2,X3,X4) -> mark(U201(X1,X2,X3,X4))
       U202(mark(X1),X2) -> mark(U202(X1,X2))
       U21(mark(X1),X2) -> mark(U21(X1,X2))
       U211(mark(X1),X2) -> mark(U211(X1,X2))
       U221(mark(X1),X2,X3) -> mark(U221(X1,X2,X3))
       fst(mark(X)) -> mark(fst(X))
       U31(mark(X1),X2) -> mark(U31(X1,X2))
       U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
       U42(mark(X1),X2) -> mark(U42(X1,X2))
       U43(mark(X)) -> mark(U43(X))
       U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
       U52(mark(X1),X2) -> mark(U52(X1,X2))
       U53(mark(X)) -> mark(U53(X))
       U61(mark(X1),X2) -> mark(U61(X1,X2))
       U62(mark(X)) -> mark(U62(X))
       U71(mark(X1),X2) -> mark(U71(X1,X2))
       U72(mark(X)) -> mark(U72(X))
       U81(mark(X1),X2) -> mark(U81(X1,X2))
       U82(mark(X)) -> mark(U82(X))
       U91(mark(X1),X2) -> mark(U91(X1,X2))
       U92(mark(X)) -> mark(U92(X))
       and(mark(X1),X2) -> mark(and(X1,X2))
       tail(mark(X)) -> mark(tail(X))
       take(mark(X1),X2) -> mark(take(X1,X2))
       take(X1,mark(X2)) -> mark(take(X1,X2))
       sel(mark(X1),X2) -> mark(sel(X1,X2))
       sel(X1,mark(X2)) -> mark(sel(X1,X2))
       proper(U101(X1,X2,X3)) -> U101(proper(X1),proper(X2),proper(X3))
       proper(tt()) -> ok(tt())
       proper(U102(X1,X2)) -> U102(proper(X1),proper(X2))
       proper(isNatural(X)) -> isNatural(proper(X))
       proper(U103(X)) -> U103(proper(X))
       proper(isLNat(X)) -> isLNat(proper(X))
       proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
       proper(snd(X)) -> snd(proper(X))
       proper(splitAt(X1,X2)) -> splitAt(proper(X1),proper(X2))
       proper(U111(X1,X2)) -> U111(proper(X1),proper(X2))
       proper(U112(X)) -> U112(proper(X))
       proper(U121(X1,X2)) -> U121(proper(X1),proper(X2))
       proper(U122(X)) -> U122(proper(X))
       proper(U131(X1,X2,X3)) -> U131(proper(X1),proper(X2),proper(X3))
       proper(U132(X1,X2)) -> U132(proper(X1),proper(X2))
       proper(U133(X)) -> U133(proper(X))
       proper(U141(X1,X2,X3)) -> U141(proper(X1),proper(X2),proper(X3))
       proper(U142(X1,X2)) -> U142(proper(X1),proper(X2))
       proper(U143(X)) -> U143(proper(X))
       proper(U151(X1,X2,X3)) -> U151(proper(X1),proper(X2),proper(X3))
       proper(U152(X1,X2)) -> U152(proper(X1),proper(X2))
       proper(U153(X)) -> U153(proper(X))
       proper(U161(X1,X2)) -> U161(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(natsFrom(X)) -> natsFrom(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(U171(X1,X2,X3)) -> U171(proper(X1),proper(X2),proper(X3))
       proper(head(X)) -> head(proper(X))
       proper(afterNth(X1,X2)) -> afterNth(proper(X1),proper(X2))
       proper(U181(X1,X2)) -> U181(proper(X1),proper(X2))
       proper(U191(X1,X2)) -> U191(proper(X1),proper(X2))
       proper(pair(X1,X2)) -> pair(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(U201(X1,X2,X3,X4)) -> U201(proper(X1),proper(X2),proper(X3),proper(X4))
       proper(U202(X1,X2)) -> U202(proper(X1),proper(X2))
       proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
       proper(U211(X1,X2)) -> U211(proper(X1),proper(X2))
       proper(U221(X1,X2,X3)) -> U221(proper(X1),proper(X2),proper(X3))
       proper(fst(X)) -> fst(proper(X))
       proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
       proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
       proper(U42(X1,X2)) -> U42(proper(X1),proper(X2))
       proper(U43(X)) -> U43(proper(X))
       proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3))
       proper(U52(X1,X2)) -> U52(proper(X1),proper(X2))
       proper(U53(X)) -> U53(proper(X))
       proper(U61(X1,X2)) -> U61(proper(X1),proper(X2))
       proper(U62(X)) -> U62(proper(X))
       proper(isPLNat(X)) -> isPLNat(proper(X))
       proper(U71(X1,X2)) -> U71(proper(X1),proper(X2))
       proper(U72(X)) -> U72(proper(X))
       proper(U81(X1,X2)) -> U81(proper(X1),proper(X2))
       proper(U82(X)) -> U82(proper(X))
       proper(U91(X1,X2)) -> U91(proper(X1),proper(X2))
       proper(U92(X)) -> U92(proper(X))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(isNaturalKind(X)) -> isNaturalKind(proper(X))
       proper(isLNatKind(X)) -> isLNatKind(proper(X))
       proper(isPLNatKind(X)) -> isPLNatKind(proper(X))
       proper(tail(X)) -> tail(proper(X))
       proper(take(X1,X2)) -> take(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(sel(X1,X2)) -> sel(proper(X1),proper(X2))
       U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
       U102(ok(X1),ok(X2)) -> ok(U102(X1,X2))
       isNatural(ok(X)) -> ok(isNatural(X))
       U103(ok(X)) -> ok(U103(X))
       isLNat(ok(X)) -> ok(isLNat(X))
       U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
       snd(ok(X)) -> ok(snd(X))
       splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2))
       U111(ok(X1),ok(X2)) -> ok(U111(X1,X2))
       U112(ok(X)) -> ok(U112(X))
       U121(ok(X1),ok(X2)) -> ok(U121(X1,X2))
       U122(ok(X)) -> ok(U122(X))
       U131(ok(X1),ok(X2),ok(X3)) -> ok(U131(X1,X2,X3))
       U132(ok(X1),ok(X2)) -> ok(U132(X1,X2))
       U133(ok(X)) -> ok(U133(X))
       U141(ok(X1),ok(X2),ok(X3)) -> ok(U141(X1,X2,X3))
       U142(ok(X1),ok(X2)) -> ok(U142(X1,X2))
       U143(ok(X)) -> ok(U143(X))
       U151(ok(X1),ok(X2),ok(X3)) -> ok(U151(X1,X2,X3))
       U152(ok(X1),ok(X2)) -> ok(U152(X1,X2))
       U153(ok(X)) -> ok(U153(X))
       U161(ok(X1),ok(X2)) -> ok(U161(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       natsFrom(ok(X)) -> ok(natsFrom(X))
       s(ok(X)) -> ok(s(X))
       U171(ok(X1),ok(X2),ok(X3)) -> ok(U171(X1,X2,X3))
       head(ok(X)) -> ok(head(X))
       afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2))
       U181(ok(X1),ok(X2)) -> ok(U181(X1,X2))
       U191(ok(X1),ok(X2)) -> ok(U191(X1,X2))
       pair(ok(X1),ok(X2)) -> ok(pair(X1,X2))
       U201(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(U201(X1,X2,X3,X4))
       U202(ok(X1),ok(X2)) -> ok(U202(X1,X2))
       U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
       U211(ok(X1),ok(X2)) -> ok(U211(X1,X2))
       U221(ok(X1),ok(X2),ok(X3)) -> ok(U221(X1,X2,X3))
       fst(ok(X)) -> ok(fst(X))
       U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
       U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
       U42(ok(X1),ok(X2)) -> ok(U42(X1,X2))
       U43(ok(X)) -> ok(U43(X))
       U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
       U52(ok(X1),ok(X2)) -> ok(U52(X1,X2))
       U53(ok(X)) -> ok(U53(X))
       U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
       U62(ok(X)) -> ok(U62(X))
       isPLNat(ok(X)) -> ok(isPLNat(X))
       U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
       U72(ok(X)) -> ok(U72(X))
       U81(ok(X1),ok(X2)) -> ok(U81(X1,X2))
       U82(ok(X)) -> ok(U82(X))
       U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
       U92(ok(X)) -> ok(U92(X))
       and(ok(X1),ok(X2)) -> ok(and(X1,X2))
       isNaturalKind(ok(X)) -> ok(isNaturalKind(X))
       isLNatKind(ok(X)) -> ok(isLNatKind(X))
       isPLNatKind(ok(X)) -> ok(isPLNatKind(X))
       tail(ok(X)) -> ok(tail(X))
       take(ok(X1),ok(X2)) -> ok(take(X1,X2))
       sel(ok(X1),ok(X2)) -> ok(sel(X1,X2))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open