(VAR V1 V2 N XS Y X YS ZS X1 X2 X3 X4)
(RULES 
a__U101(tt,V1,V2) -> a__U102(a__isNatural(V1),V2)
a__U102(tt,V2) -> a__U103(a__isLNat(V2))
a__U103(tt) -> tt
a__U11(tt,N,XS) -> a__snd(a__splitAt(mark(N),mark(XS)))
a__U111(tt,V1) -> a__U112(a__isLNat(V1))
a__U112(tt) -> tt
a__U121(tt,V1) -> a__U122(a__isNatural(V1))
a__U122(tt) -> tt
a__U131(tt,V1,V2) -> a__U132(a__isNatural(V1),V2)
a__U132(tt,V2) -> a__U133(a__isLNat(V2))
a__U133(tt) -> tt
a__U141(tt,V1,V2) -> a__U142(a__isLNat(V1),V2)
a__U142(tt,V2) -> a__U143(a__isLNat(V2))
a__U143(tt) -> tt
a__U151(tt,V1,V2) -> a__U152(a__isNatural(V1),V2)
a__U152(tt,V2) -> a__U153(a__isLNat(V2))
a__U153(tt) -> tt
a__U161(tt,N) -> cons(mark(N),natsFrom(s(N)))
a__U171(tt,N,XS) -> a__head(a__afterNth(mark(N),mark(XS)))
a__U181(tt,Y) -> mark(Y)
a__U191(tt,XS) -> pair(nil,mark(XS))
a__U201(tt,N,X,XS) -> a__U202(a__splitAt(mark(N),mark(XS)),X)
a__U202(pair(YS,ZS),X) -> pair(cons(mark(X),YS),mark(ZS))
a__U21(tt,X) -> mark(X)
a__U211(tt,XS) -> mark(XS)
a__U221(tt,N,XS) -> a__fst(a__splitAt(mark(N),mark(XS)))
a__U31(tt,N) -> mark(N)
a__U41(tt,V1,V2) -> a__U42(a__isNatural(V1),V2)
a__U42(tt,V2) -> a__U43(a__isLNat(V2))
a__U43(tt) -> tt
a__U51(tt,V1,V2) -> a__U52(a__isNatural(V1),V2)
a__U52(tt,V2) -> a__U53(a__isLNat(V2))
a__U53(tt) -> tt
a__U61(tt,V1) -> a__U62(a__isPLNat(V1))
a__U62(tt) -> tt
a__U71(tt,V1) -> a__U72(a__isNatural(V1))
a__U72(tt) -> tt
a__U81(tt,V1) -> a__U82(a__isPLNat(V1))
a__U82(tt) -> tt
a__U91(tt,V1) -> a__U92(a__isLNat(V1))
a__U92(tt) -> tt
a__afterNth(N,XS) -> a__U11(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
a__and(tt,X) -> mark(X)
a__fst(pair(X,Y)) -> a__U21(a__and(a__and(a__isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),X)
a__head(cons(N,XS)) -> a__U31(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N)
a__isLNat(nil) -> tt
a__isLNat(afterNth(V1,V2)) -> a__U41(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNat(cons(V1,V2)) -> a__U51(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNat(fst(V1)) -> a__U61(a__isPLNatKind(V1),V1)
a__isLNat(natsFrom(V1)) -> a__U71(a__isNaturalKind(V1),V1)
a__isLNat(snd(V1)) -> a__U81(a__isPLNatKind(V1),V1)
a__isLNat(tail(V1)) -> a__U91(a__isLNatKind(V1),V1)
a__isLNat(take(V1,V2)) -> a__U101(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isLNatKind(nil) -> tt
a__isLNatKind(afterNth(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isLNatKind(cons(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isLNatKind(fst(V1)) -> a__isPLNatKind(V1)
a__isLNatKind(natsFrom(V1)) -> a__isNaturalKind(V1)
a__isLNatKind(snd(V1)) -> a__isPLNatKind(V1)
a__isLNatKind(tail(V1)) -> a__isLNatKind(V1)
a__isLNatKind(take(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isNatural(0) -> tt
a__isNatural(head(V1)) -> a__U111(a__isLNatKind(V1),V1)
a__isNatural(s(V1)) -> a__U121(a__isNaturalKind(V1),V1)
a__isNatural(sel(V1,V2)) -> a__U131(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isNaturalKind(0) -> tt
a__isNaturalKind(head(V1)) -> a__isLNatKind(V1)
a__isNaturalKind(s(V1)) -> a__isNaturalKind(V1)
a__isNaturalKind(sel(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__isPLNat(pair(V1,V2)) -> a__U141(a__and(a__isLNatKind(V1),isLNatKind(V2)),V1,V2)
a__isPLNat(splitAt(V1,V2)) -> a__U151(a__and(a__isNaturalKind(V1),isLNatKind(V2)),V1,V2)
a__isPLNatKind(pair(V1,V2)) -> a__and(a__isLNatKind(V1),isLNatKind(V2))
a__isPLNatKind(splitAt(V1,V2)) -> a__and(a__isNaturalKind(V1),isLNatKind(V2))
a__natsFrom(N) -> a__U161(a__and(a__isNatural(N),isNaturalKind(N)),N)
a__sel(N,XS) -> a__U171(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
a__snd(pair(X,Y)) -> a__U181(a__and(a__and(a__isLNat(X),isLNatKind(X)),and(isLNat(Y),isLNatKind(Y))),Y)
a__splitAt(0,XS) -> a__U191(a__and(a__isLNat(XS),isLNatKind(XS)),XS)
a__splitAt(s(N),cons(X,XS)) -> a__U201(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(and(isNatural(X),isNaturalKind(X)),and(isLNat(XS),isLNatKind(XS)))),N,X,XS)
a__tail(cons(N,XS)) -> a__U211(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),XS)
a__take(N,XS) -> a__U221(a__and(a__and(a__isNatural(N),isNaturalKind(N)),and(isLNat(XS),isLNatKind(XS))),N,XS)
mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
mark(U102(X1,X2)) -> a__U102(mark(X1),X2)
mark(isNatural(X)) -> a__isNatural(X)
mark(U103(X)) -> a__U103(mark(X))
mark(isLNat(X)) -> a__isLNat(X)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(snd(X)) -> a__snd(mark(X))
mark(splitAt(X1,X2)) -> a__splitAt(mark(X1),mark(X2))
mark(U111(X1,X2)) -> a__U111(mark(X1),X2)
mark(U112(X)) -> a__U112(mark(X))
mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
mark(U122(X)) -> a__U122(mark(X))
mark(U131(X1,X2,X3)) -> a__U131(mark(X1),X2,X3)
mark(U132(X1,X2)) -> a__U132(mark(X1),X2)
mark(U133(X)) -> a__U133(mark(X))
mark(U141(X1,X2,X3)) -> a__U141(mark(X1),X2,X3)
mark(U142(X1,X2)) -> a__U142(mark(X1),X2)
mark(U143(X)) -> a__U143(mark(X))
mark(U151(X1,X2,X3)) -> a__U151(mark(X1),X2,X3)
mark(U152(X1,X2)) -> a__U152(mark(X1),X2)
mark(U153(X)) -> a__U153(mark(X))
mark(U161(X1,X2)) -> a__U161(mark(X1),X2)
mark(natsFrom(X)) -> a__natsFrom(mark(X))
mark(U171(X1,X2,X3)) -> a__U171(mark(X1),X2,X3)
mark(head(X)) -> a__head(mark(X))
mark(afterNth(X1,X2)) -> a__afterNth(mark(X1),mark(X2))
mark(U181(X1,X2)) -> a__U181(mark(X1),X2)
mark(U191(X1,X2)) -> a__U191(mark(X1),X2)
mark(U201(X1,X2,X3,X4)) -> a__U201(mark(X1),X2,X3,X4)
mark(U202(X1,X2)) -> a__U202(mark(X1),X2)
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U211(X1,X2)) -> a__U211(mark(X1),X2)
mark(U221(X1,X2,X3)) -> a__U221(mark(X1),X2,X3)
mark(fst(X)) -> a__fst(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
mark(U42(X1,X2)) -> a__U42(mark(X1),X2)
mark(U43(X)) -> a__U43(mark(X))
mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U53(X)) -> a__U53(mark(X))
mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
mark(U62(X)) -> a__U62(mark(X))
mark(isPLNat(X)) -> a__isPLNat(X)
mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
mark(U72(X)) -> a__U72(mark(X))
mark(U81(X1,X2)) -> a__U81(mark(X1),X2)
mark(U82(X)) -> a__U82(mark(X))
mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
mark(U92(X)) -> a__U92(mark(X))
mark(and(X1,X2)) -> a__and(mark(X1),X2)
mark(isNaturalKind(X)) -> a__isNaturalKind(X)
mark(isLNatKind(X)) -> a__isLNatKind(X)
mark(isPLNatKind(X)) -> a__isPLNatKind(X)
mark(tail(X)) -> a__tail(mark(X))
mark(take(X1,X2)) -> a__take(mark(X1),mark(X2))
mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2))
mark(tt) -> tt
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(s(X)) -> s(mark(X))
mark(pair(X1,X2)) -> pair(mark(X1),mark(X2))
mark(nil) -> nil
mark(0) -> 0
a__U101(X1,X2,X3) -> U101(X1,X2,X3)
a__U102(X1,X2) -> U102(X1,X2)
a__isNatural(X) -> isNatural(X)
a__U103(X) -> U103(X)
a__isLNat(X) -> isLNat(X)
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__snd(X) -> snd(X)
a__splitAt(X1,X2) -> splitAt(X1,X2)
a__U111(X1,X2) -> U111(X1,X2)
a__U112(X) -> U112(X)
a__U121(X1,X2) -> U121(X1,X2)
a__U122(X) -> U122(X)
a__U131(X1,X2,X3) -> U131(X1,X2,X3)
a__U132(X1,X2) -> U132(X1,X2)
a__U133(X) -> U133(X)
a__U141(X1,X2,X3) -> U141(X1,X2,X3)
a__U142(X1,X2) -> U142(X1,X2)
a__U143(X) -> U143(X)
a__U151(X1,X2,X3) -> U151(X1,X2,X3)
a__U152(X1,X2) -> U152(X1,X2)
a__U153(X) -> U153(X)
a__U161(X1,X2) -> U161(X1,X2)
a__natsFrom(X) -> natsFrom(X)
a__U171(X1,X2,X3) -> U171(X1,X2,X3)
a__head(X) -> head(X)
a__afterNth(X1,X2) -> afterNth(X1,X2)
a__U181(X1,X2) -> U181(X1,X2)
a__U191(X1,X2) -> U191(X1,X2)
a__U201(X1,X2,X3,X4) -> U201(X1,X2,X3,X4)
a__U202(X1,X2) -> U202(X1,X2)
a__U21(X1,X2) -> U21(X1,X2)
a__U211(X1,X2) -> U211(X1,X2)
a__U221(X1,X2,X3) -> U221(X1,X2,X3)
a__fst(X) -> fst(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U41(X1,X2,X3) -> U41(X1,X2,X3)
a__U42(X1,X2) -> U42(X1,X2)
a__U43(X) -> U43(X)
a__U51(X1,X2,X3) -> U51(X1,X2,X3)
a__U52(X1,X2) -> U52(X1,X2)
a__U53(X) -> U53(X)
a__U61(X1,X2) -> U61(X1,X2)
a__U62(X) -> U62(X)
a__isPLNat(X) -> isPLNat(X)
a__U71(X1,X2) -> U71(X1,X2)
a__U72(X) -> U72(X)
a__U81(X1,X2) -> U81(X1,X2)
a__U82(X) -> U82(X)
a__U91(X1,X2) -> U91(X1,X2)
a__U92(X) -> U92(X)
a__and(X1,X2) -> and(X1,X2)
a__isNaturalKind(X) -> isNaturalKind(X)
a__isLNatKind(X) -> isLNatKind(X)
a__isPLNatKind(X) -> isPLNatKind(X)
a__tail(X) -> tail(X)
a__take(X1,X2) -> take(X1,X2)
a__sel(X1,X2) -> sel(X1,X2)
)