(VAR V1 V2 N XS Y X YS ZS X1 X2 X3 X4)
(RULES 
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))
)