Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following 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)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U101(tt, V1, V2) → A__U102(a__isNatural(V1), V2)
A__U101(tt, V1, V2) → A__ISNATURAL(V1)
A__U102(tt, V2) → A__U103(a__isLNat(V2))
A__U102(tt, V2) → A__ISLNAT(V2)
A__U11(tt, N, XS) → A__SND(a__splitAt(mark(N), mark(XS)))
A__U11(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U11(tt, N, XS) → MARK(N)
A__U11(tt, N, XS) → MARK(XS)
A__U111(tt, V1) → A__U112(a__isLNat(V1))
A__U111(tt, V1) → A__ISLNAT(V1)
A__U121(tt, V1) → A__U122(a__isNatural(V1))
A__U121(tt, V1) → A__ISNATURAL(V1)
A__U131(tt, V1, V2) → A__U132(a__isNatural(V1), V2)
A__U131(tt, V1, V2) → A__ISNATURAL(V1)
A__U132(tt, V2) → A__U133(a__isLNat(V2))
A__U132(tt, V2) → A__ISLNAT(V2)
A__U141(tt, V1, V2) → A__U142(a__isLNat(V1), V2)
A__U141(tt, V1, V2) → A__ISLNAT(V1)
A__U142(tt, V2) → A__U143(a__isLNat(V2))
A__U142(tt, V2) → A__ISLNAT(V2)
A__U151(tt, V1, V2) → A__U152(a__isNatural(V1), V2)
A__U151(tt, V1, V2) → A__ISNATURAL(V1)
A__U152(tt, V2) → A__U153(a__isLNat(V2))
A__U152(tt, V2) → A__ISLNAT(V2)
A__U161(tt, N) → MARK(N)
A__U171(tt, N, XS) → A__HEAD(a__afterNth(mark(N), mark(XS)))
A__U171(tt, N, XS) → A__AFTERNTH(mark(N), mark(XS))
A__U171(tt, N, XS) → MARK(N)
A__U171(tt, N, XS) → MARK(XS)
A__U181(tt, Y) → MARK(Y)
A__U191(tt, XS) → MARK(XS)
A__U201(tt, N, X, XS) → A__U202(a__splitAt(mark(N), mark(XS)), X)
A__U201(tt, N, X, XS) → A__SPLITAT(mark(N), mark(XS))
A__U201(tt, N, X, XS) → MARK(N)
A__U201(tt, N, X, XS) → MARK(XS)
A__U202(pair(YS, ZS), X) → MARK(X)
A__U202(pair(YS, ZS), X) → 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__U221(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U221(tt, N, XS) → MARK(N)
A__U221(tt, N, XS) → MARK(XS)
A__U31(tt, N) → MARK(N)
A__U41(tt, V1, V2) → A__U42(a__isNatural(V1), V2)
A__U41(tt, V1, V2) → A__ISNATURAL(V1)
A__U42(tt, V2) → A__U43(a__isLNat(V2))
A__U42(tt, V2) → A__ISLNAT(V2)
A__U51(tt, V1, V2) → A__U52(a__isNatural(V1), V2)
A__U51(tt, V1, V2) → A__ISNATURAL(V1)
A__U52(tt, V2) → A__U53(a__isLNat(V2))
A__U52(tt, V2) → A__ISLNAT(V2)
A__U61(tt, V1) → A__U62(a__isPLNat(V1))
A__U61(tt, V1) → A__ISPLNAT(V1)
A__U71(tt, V1) → A__U72(a__isNatural(V1))
A__U71(tt, V1) → A__ISNATURAL(V1)
A__U81(tt, V1) → A__U82(a__isPLNat(V1))
A__U81(tt, V1) → A__ISPLNAT(V1)
A__U91(tt, V1) → A__U92(a__isLNat(V1))
A__U91(tt, V1) → A__ISLNAT(V1)
A__AFTERNTH(N, XS) → A__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
A__AFTERNTH(N, XS) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__AFTERNTH(N, XS) → A__AND(a__isNatural(N), isNaturalKind(N))
A__AFTERNTH(N, XS) → A__ISNATURAL(N)
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__FST(pair(X, Y)) → A__AND(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y)))
A__FST(pair(X, Y)) → A__AND(a__isLNat(X), isLNatKind(X))
A__FST(pair(X, Y)) → A__ISLNAT(X)
A__HEAD(cons(N, XS)) → A__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N)
A__HEAD(cons(N, XS)) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__HEAD(cons(N, XS)) → A__AND(a__isNatural(N), isNaturalKind(N))
A__HEAD(cons(N, XS)) → A__ISNATURAL(N)
A__ISLNAT(afterNth(V1, V2)) → A__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__ISLNAT(afterNth(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNAT(afterNth(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNAT(cons(V1, V2)) → A__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__ISLNAT(cons(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNAT(cons(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNAT(fst(V1)) → A__U61(a__isPLNatKind(V1), V1)
A__ISLNAT(fst(V1)) → A__ISPLNATKIND(V1)
A__ISLNAT(natsFrom(V1)) → A__U71(a__isNaturalKind(V1), V1)
A__ISLNAT(natsFrom(V1)) → A__ISNATURALKIND(V1)
A__ISLNAT(snd(V1)) → A__U81(a__isPLNatKind(V1), V1)
A__ISLNAT(snd(V1)) → A__ISPLNATKIND(V1)
A__ISLNAT(tail(V1)) → A__U91(a__isLNatKind(V1), V1)
A__ISLNAT(tail(V1)) → A__ISLNATKIND(V1)
A__ISLNAT(take(V1, V2)) → A__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__ISLNAT(take(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNAT(take(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNATKIND(afterNth(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNATKIND(afterNth(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNATKIND(cons(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNATKIND(cons(V1, V2)) → A__ISNATURALKIND(V1)
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__ISLNATKIND(take(V1, V2)) → A__ISNATURALKIND(V1)
A__ISNATURAL(head(V1)) → A__U111(a__isLNatKind(V1), V1)
A__ISNATURAL(head(V1)) → A__ISLNATKIND(V1)
A__ISNATURAL(s(V1)) → A__U121(a__isNaturalKind(V1), V1)
A__ISNATURAL(s(V1)) → A__ISNATURALKIND(V1)
A__ISNATURAL(sel(V1, V2)) → A__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__ISNATURAL(sel(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISNATURAL(sel(V1, V2)) → A__ISNATURALKIND(V1)
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__ISNATURALKIND(sel(V1, V2)) → A__ISNATURALKIND(V1)
A__ISPLNAT(pair(V1, V2)) → A__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2)
A__ISPLNAT(pair(V1, V2)) → A__AND(a__isLNatKind(V1), isLNatKind(V2))
A__ISPLNAT(pair(V1, V2)) → A__ISLNATKIND(V1)
A__ISPLNAT(splitAt(V1, V2)) → A__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__ISPLNAT(splitAt(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISPLNAT(splitAt(V1, V2)) → A__ISNATURALKIND(V1)
A__ISPLNATKIND(pair(V1, V2)) → A__AND(a__isLNatKind(V1), isLNatKind(V2))
A__ISPLNATKIND(pair(V1, V2)) → A__ISLNATKIND(V1)
A__ISPLNATKIND(splitAt(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISPLNATKIND(splitAt(V1, V2)) → A__ISNATURALKIND(V1)
A__NATSFROM(N) → A__U161(a__and(a__isNatural(N), isNaturalKind(N)), N)
A__NATSFROM(N) → A__AND(a__isNatural(N), isNaturalKind(N))
A__NATSFROM(N) → A__ISNATURAL(N)
A__SEL(N, XS) → A__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
A__SEL(N, XS) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__SEL(N, XS) → A__AND(a__isNatural(N), isNaturalKind(N))
A__SEL(N, XS) → A__ISNATURAL(N)
A__SND(pair(X, Y)) → A__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y)
A__SND(pair(X, Y)) → A__AND(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y)))
A__SND(pair(X, Y)) → A__AND(a__isLNat(X), isLNatKind(X))
A__SND(pair(X, Y)) → A__ISLNAT(X)
A__SPLITAT(0, XS) → A__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS)
A__SPLITAT(0, XS) → A__AND(a__isLNat(XS), isLNatKind(XS))
A__SPLITAT(0, XS) → A__ISLNAT(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__SPLITAT(s(N), cons(X, XS)) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS))))
A__SPLITAT(s(N), cons(X, XS)) → A__AND(a__isNatural(N), isNaturalKind(N))
A__SPLITAT(s(N), cons(X, XS)) → A__ISNATURAL(N)
A__TAIL(cons(N, XS)) → A__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS)
A__TAIL(cons(N, XS)) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__TAIL(cons(N, XS)) → A__AND(a__isNatural(N), isNaturalKind(N))
A__TAIL(cons(N, XS)) → A__ISNATURAL(N)
A__TAKE(N, XS) → A__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
A__TAKE(N, XS) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__TAKE(N, XS) → A__AND(a__isNatural(N), isNaturalKind(N))
A__TAKE(N, XS) → A__ISNATURAL(N)
MARK(U101(X1, X2, X3)) → A__U101(mark(X1), X2, X3)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2)) → A__U102(mark(X1), X2)
MARK(U102(X1, X2)) → MARK(X1)
MARK(isNatural(X)) → A__ISNATURAL(X)
MARK(U103(X)) → A__U103(mark(X))
MARK(U103(X)) → MARK(X)
MARK(isLNat(X)) → A__ISLNAT(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → A__SND(mark(X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → A__SPLITAT(mark(X1), mark(X2))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U111(X1, X2)) → A__U111(mark(X1), X2)
MARK(U111(X1, X2)) → MARK(X1)
MARK(U112(X)) → A__U112(mark(X))
MARK(U112(X)) → MARK(X)
MARK(U121(X1, X2)) → A__U121(mark(X1), X2)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U122(X)) → A__U122(mark(X))
MARK(U122(X)) → MARK(X)
MARK(U131(X1, X2, X3)) → A__U131(mark(X1), X2, X3)
MARK(U131(X1, X2, X3)) → MARK(X1)
MARK(U132(X1, X2)) → A__U132(mark(X1), X2)
MARK(U132(X1, X2)) → MARK(X1)
MARK(U133(X)) → A__U133(mark(X))
MARK(U133(X)) → MARK(X)
MARK(U141(X1, X2, X3)) → A__U141(mark(X1), X2, X3)
MARK(U141(X1, X2, X3)) → MARK(X1)
MARK(U142(X1, X2)) → A__U142(mark(X1), X2)
MARK(U142(X1, X2)) → MARK(X1)
MARK(U143(X)) → A__U143(mark(X))
MARK(U143(X)) → MARK(X)
MARK(U151(X1, X2, X3)) → A__U151(mark(X1), X2, X3)
MARK(U151(X1, X2, X3)) → MARK(X1)
MARK(U152(X1, X2)) → A__U152(mark(X1), X2)
MARK(U152(X1, X2)) → MARK(X1)
MARK(U153(X)) → A__U153(mark(X))
MARK(U153(X)) → MARK(X)
MARK(U161(X1, X2)) → A__U161(mark(X1), X2)
MARK(U161(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → A__NATSFROM(mark(X))
MARK(natsFrom(X)) → MARK(X)
MARK(U171(X1, X2, X3)) → A__U171(mark(X1), X2, X3)
MARK(U171(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → A__HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → A__AFTERNTH(mark(X1), mark(X2))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U181(X1, X2)) → A__U181(mark(X1), X2)
MARK(U181(X1, X2)) → MARK(X1)
MARK(U191(X1, X2)) → A__U191(mark(X1), X2)
MARK(U191(X1, X2)) → MARK(X1)
MARK(U201(X1, X2, X3, X4)) → A__U201(mark(X1), X2, X3, X4)
MARK(U201(X1, X2, X3, X4)) → MARK(X1)
MARK(U202(X1, X2)) → A__U202(mark(X1), X2)
MARK(U202(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U211(X1, X2)) → A__U211(mark(X1), X2)
MARK(U211(X1, X2)) → MARK(X1)
MARK(U221(X1, X2, X3)) → A__U221(mark(X1), X2, X3)
MARK(U221(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → A__FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → A__U43(mark(X))
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U53(X)) → A__U53(mark(X))
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → A__U62(mark(X))
MARK(U62(X)) → MARK(X)
MARK(isPLNat(X)) → A__ISPLNAT(X)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → A__U72(mark(X))
MARK(U72(X)) → MARK(X)
MARK(U81(X1, X2)) → A__U81(mark(X1), X2)
MARK(U81(X1, X2)) → MARK(X1)
MARK(U82(X)) → A__U82(mark(X))
MARK(U82(X)) → MARK(X)
MARK(U91(X1, X2)) → A__U91(mark(X1), X2)
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → A__U92(mark(X))
MARK(U92(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
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(tail(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)

The TRS R consists of the following 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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 24 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U102(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(afterNth(V1, V2)) → A__U41(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatural(V1), V2)
A__U42(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(afterNth(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U101(X1, X2, X3)) → A__U101(mark(X1), X2, X3)
A__U101(tt, V1, V2) → A__U102(a__isNatural(V1), V2)
A__U101(tt, V1, V2) → A__ISNATURAL(V1)
A__ISNATURAL(head(V1)) → A__U111(a__isLNatKind(V1), V1)
A__U111(tt, V1) → A__ISLNAT(V1)
A__ISLNAT(afterNth(V1, V2)) → A__ISNATURALKIND(V1)
A__ISNATURALKIND(head(V1)) → A__ISLNATKIND(V1)
A__ISLNATKIND(afterNth(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNATKIND(afterNth(V1, V2)) → A__ISNATURALKIND(V1)
A__ISNATURALKIND(s(V1)) → A__ISNATURALKIND(V1)
A__ISNATURALKIND(sel(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISNATURALKIND(sel(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNATKIND(cons(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNATKIND(cons(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNATKIND(fst(V1)) → A__ISPLNATKIND(V1)
A__ISPLNATKIND(pair(V1, V2)) → A__AND(a__isLNatKind(V1), isLNatKind(V2))
A__ISPLNATKIND(pair(V1, V2)) → A__ISLNATKIND(V1)
A__ISLNATKIND(natsFrom(V1)) → A__ISNATURALKIND(V1)
A__ISLNATKIND(snd(V1)) → A__ISPLNATKIND(V1)
A__ISPLNATKIND(splitAt(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISPLNATKIND(splitAt(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNATKIND(tail(V1)) → A__ISLNATKIND(V1)
A__ISLNATKIND(take(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNATKIND(take(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNAT(cons(V1, V2)) → A__U51(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__U51(tt, V1, V2) → A__U52(a__isNatural(V1), V2)
A__U52(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(cons(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNAT(cons(V1, V2)) → A__ISNATURALKIND(V1)
A__ISLNAT(fst(V1)) → A__U61(a__isPLNatKind(V1), V1)
A__U61(tt, V1) → A__ISPLNAT(V1)
A__ISPLNAT(pair(V1, V2)) → A__U141(a__and(a__isLNatKind(V1), isLNatKind(V2)), V1, V2)
A__U141(tt, V1, V2) → A__U142(a__isLNat(V1), V2)
A__U142(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(fst(V1)) → A__ISPLNATKIND(V1)
A__ISLNAT(natsFrom(V1)) → A__U71(a__isNaturalKind(V1), V1)
A__U71(tt, V1) → A__ISNATURAL(V1)
A__ISNATURAL(head(V1)) → A__ISLNATKIND(V1)
A__ISNATURAL(s(V1)) → A__U121(a__isNaturalKind(V1), V1)
A__U121(tt, V1) → A__ISNATURAL(V1)
A__ISNATURAL(s(V1)) → A__ISNATURALKIND(V1)
A__ISNATURAL(sel(V1, V2)) → A__U131(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__U131(tt, V1, V2) → A__U132(a__isNatural(V1), V2)
A__U132(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(natsFrom(V1)) → A__ISNATURALKIND(V1)
A__ISLNAT(snd(V1)) → A__U81(a__isPLNatKind(V1), V1)
A__U81(tt, V1) → A__ISPLNAT(V1)
A__ISPLNAT(pair(V1, V2)) → A__AND(a__isLNatKind(V1), isLNatKind(V2))
A__ISPLNAT(pair(V1, V2)) → A__ISLNATKIND(V1)
A__ISPLNAT(splitAt(V1, V2)) → A__U151(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__U151(tt, V1, V2) → A__U152(a__isNatural(V1), V2)
A__U152(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(snd(V1)) → A__ISPLNATKIND(V1)
A__ISLNAT(tail(V1)) → A__U91(a__isLNatKind(V1), V1)
A__U91(tt, V1) → A__ISLNAT(V1)
A__ISLNAT(tail(V1)) → A__ISLNATKIND(V1)
A__ISLNAT(take(V1, V2)) → A__U101(a__and(a__isNaturalKind(V1), isLNatKind(V2)), V1, V2)
A__ISLNAT(take(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISLNAT(take(V1, V2)) → A__ISNATURALKIND(V1)
A__U151(tt, V1, V2) → A__ISNATURAL(V1)
A__ISNATURAL(sel(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISNATURAL(sel(V1, V2)) → A__ISNATURALKIND(V1)
A__ISPLNAT(splitAt(V1, V2)) → A__AND(a__isNaturalKind(V1), isLNatKind(V2))
A__ISPLNAT(splitAt(V1, V2)) → A__ISNATURALKIND(V1)
A__U131(tt, V1, V2) → A__ISNATURAL(V1)
A__U141(tt, V1, V2) → A__ISLNAT(V1)
A__U51(tt, V1, V2) → A__ISNATURAL(V1)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2)) → A__U102(mark(X1), X2)
MARK(U102(X1, X2)) → MARK(X1)
MARK(isNatural(X)) → A__ISNATURAL(X)
MARK(U103(X)) → MARK(X)
MARK(isLNat(X)) → A__ISLNAT(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, N, XS) → A__SND(a__splitAt(mark(N), mark(XS)))
A__SND(pair(X, Y)) → A__U181(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y)
A__U181(tt, Y) → MARK(Y)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → A__SND(mark(X))
A__SND(pair(X, Y)) → A__AND(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y)))
A__SND(pair(X, Y)) → A__AND(a__isLNat(X), isLNatKind(X))
A__SND(pair(X, Y)) → A__ISLNAT(X)
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → A__SPLITAT(mark(X1), mark(X2))
A__SPLITAT(0, XS) → A__U191(a__and(a__isLNat(XS), isLNatKind(XS)), XS)
A__U191(tt, XS) → MARK(XS)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U111(X1, X2)) → A__U111(mark(X1), X2)
MARK(U111(X1, X2)) → MARK(X1)
MARK(U112(X)) → MARK(X)
MARK(U121(X1, X2)) → A__U121(mark(X1), X2)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U122(X)) → MARK(X)
MARK(U131(X1, X2, X3)) → A__U131(mark(X1), X2, X3)
MARK(U131(X1, X2, X3)) → MARK(X1)
MARK(U132(X1, X2)) → A__U132(mark(X1), X2)
MARK(U132(X1, X2)) → MARK(X1)
MARK(U133(X)) → MARK(X)
MARK(U141(X1, X2, X3)) → A__U141(mark(X1), X2, X3)
MARK(U141(X1, X2, X3)) → MARK(X1)
MARK(U142(X1, X2)) → A__U142(mark(X1), X2)
MARK(U142(X1, X2)) → MARK(X1)
MARK(U143(X)) → MARK(X)
MARK(U151(X1, X2, X3)) → A__U151(mark(X1), X2, X3)
MARK(U151(X1, X2, X3)) → MARK(X1)
MARK(U152(X1, X2)) → A__U152(mark(X1), X2)
MARK(U152(X1, X2)) → MARK(X1)
MARK(U153(X)) → MARK(X)
MARK(U161(X1, X2)) → A__U161(mark(X1), X2)
A__U161(tt, N) → MARK(N)
MARK(U161(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → A__NATSFROM(mark(X))
A__NATSFROM(N) → A__U161(a__and(a__isNatural(N), isNaturalKind(N)), N)
A__NATSFROM(N) → A__AND(a__isNatural(N), isNaturalKind(N))
A__NATSFROM(N) → A__ISNATURAL(N)
MARK(natsFrom(X)) → MARK(X)
MARK(U171(X1, X2, X3)) → A__U171(mark(X1), X2, X3)
A__U171(tt, N, XS) → A__HEAD(a__afterNth(mark(N), mark(XS)))
A__HEAD(cons(N, XS)) → A__U31(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N)
A__U31(tt, N) → MARK(N)
MARK(U171(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → A__HEAD(mark(X))
A__HEAD(cons(N, XS)) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__HEAD(cons(N, XS)) → A__AND(a__isNatural(N), isNaturalKind(N))
A__HEAD(cons(N, XS)) → A__ISNATURAL(N)
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → A__AFTERNTH(mark(X1), mark(X2))
A__AFTERNTH(N, XS) → A__U11(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
A__U11(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__SPLITAT(0, XS) → A__AND(a__isLNat(XS), isLNatKind(XS))
A__SPLITAT(0, XS) → A__ISLNAT(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__U201(tt, N, X, XS) → A__U202(a__splitAt(mark(N), mark(XS)), X)
A__U202(pair(YS, ZS), X) → MARK(X)
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U181(X1, X2)) → A__U181(mark(X1), X2)
MARK(U181(X1, X2)) → MARK(X1)
MARK(U191(X1, X2)) → A__U191(mark(X1), X2)
MARK(U191(X1, X2)) → MARK(X1)
MARK(U201(X1, X2, X3, X4)) → A__U201(mark(X1), X2, X3, X4)
A__U201(tt, N, X, XS) → A__SPLITAT(mark(N), mark(XS))
A__SPLITAT(s(N), cons(X, XS)) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS))))
A__SPLITAT(s(N), cons(X, XS)) → A__AND(a__isNatural(N), isNaturalKind(N))
A__SPLITAT(s(N), cons(X, XS)) → A__ISNATURAL(N)
A__U201(tt, N, X, XS) → MARK(N)
MARK(U201(X1, X2, X3, X4)) → MARK(X1)
MARK(U202(X1, X2)) → A__U202(mark(X1), X2)
A__U202(pair(YS, ZS), X) → MARK(ZS)
MARK(U202(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U21(tt, X) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U211(X1, X2)) → A__U211(mark(X1), X2)
A__U211(tt, XS) → MARK(XS)
MARK(U211(X1, X2)) → MARK(X1)
MARK(U221(X1, X2, X3)) → A__U221(mark(X1), X2, X3)
A__U221(tt, N, XS) → A__FST(a__splitAt(mark(N), mark(XS)))
A__FST(pair(X, Y)) → A__U21(a__and(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X)
A__FST(pair(X, Y)) → A__AND(a__and(a__isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y)))
A__FST(pair(X, Y)) → A__AND(a__isLNat(X), isLNatKind(X))
A__FST(pair(X, Y)) → A__ISLNAT(X)
A__U221(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U221(tt, N, XS) → MARK(N)
MARK(U221(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → A__FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U41(tt, V1, V2) → A__ISNATURAL(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(isPLNat(X)) → A__ISPLNAT(X)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U81(X1, X2)) → A__U81(mark(X1), X2)
MARK(U81(X1, X2)) → MARK(X1)
MARK(U82(X)) → MARK(X)
MARK(U91(X1, X2)) → A__U91(mark(X1), X2)
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
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))
A__TAIL(cons(N, XS)) → A__U211(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS)
A__TAIL(cons(N, XS)) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__TAIL(cons(N, XS)) → A__AND(a__isNatural(N), isNaturalKind(N))
A__TAIL(cons(N, XS)) → A__ISNATURAL(N)
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(N, XS) → A__U221(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
A__U221(tt, N, XS) → MARK(XS)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(N, XS) → A__U171(a__and(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)
A__U171(tt, N, XS) → A__AFTERNTH(mark(N), mark(XS))
A__AFTERNTH(N, XS) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__AFTERNTH(N, XS) → A__AND(a__isNatural(N), isNaturalKind(N))
A__AFTERNTH(N, XS) → A__ISNATURAL(N)
A__U171(tt, N, XS) → MARK(N)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
A__U171(tt, N, XS) → MARK(XS)
A__SEL(N, XS) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__SEL(N, XS) → A__AND(a__isNatural(N), isNaturalKind(N))
A__SEL(N, XS) → A__ISNATURAL(N)
A__TAKE(N, XS) → A__AND(a__and(a__isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS)))
A__TAKE(N, XS) → A__AND(a__isNatural(N), isNaturalKind(N))
A__TAKE(N, XS) → A__ISNATURAL(N)
A__U201(tt, N, X, XS) → MARK(XS)
A__U11(tt, N, XS) → MARK(N)
A__U11(tt, N, XS) → MARK(XS)

The TRS R consists of the following 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)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.