Termination w.r.t. Q proof of /tmp/tmp3XKFzu/LISTUTILITIES_nokinds_noand

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 DependencyGraphProof (⇔)

    ↳9 TRUE

  ↳10 QDP

(0) Obligation:

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

a__U101(tt, V2) → a__U102(a__isLNat(V2))
a__U102(tt) → tt
a__U11(tt, N, XS) → a__U12(a__isLNat(XS), N, XS)
a__U111(tt) → tt
a__U12(tt, N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__U121(tt) → tt
a__U131(tt, V2) → a__U132(a__isLNat(V2))
a__U132(tt) → tt
a__U141(tt, V2) → a__U142(a__isLNat(V2))
a__U142(tt) → tt
a__U151(tt, V2) → a__U152(a__isLNat(V2))
a__U152(tt) → tt
a__U161(tt, N) → cons(mark(N), natsFrom(s(N)))
a__U171(tt, N, XS) → a__U172(a__isLNat(XS), N, XS)
a__U172(tt, N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__U181(tt, Y) → a__U182(a__isLNat(Y), Y)
a__U182(tt, Y) → mark(Y)
a__U191(tt, XS) → pair(nil, mark(XS))
a__U201(tt, N, X, XS) → a__U202(a__isNatural(X), N, X, XS)
a__U202(tt, N, X, XS) → a__U203(a__isLNat(XS), N, X, XS)
a__U203(tt, N, X, XS) → a__U204(a__splitAt(mark(N), mark(XS)), X)
a__U204(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__U21(tt, X, Y) → a__U22(a__isLNat(Y), X)
a__U211(tt, XS) → a__U212(a__isLNat(XS), XS)
a__U212(tt, XS) → mark(XS)
a__U22(tt, X) → mark(X)
a__U221(tt, N, XS) → a__U222(a__isLNat(XS), N, XS)
a__U222(tt, N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
a__U31(tt, N, XS) → a__U32(a__isLNat(XS), N)
a__U32(tt, N) → mark(N)
a__U41(tt, V2) → a__U42(a__isLNat(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isLNat(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt) → tt
a__afterNth(N, XS) → a__U11(a__isNatural(N), N, XS)
a__fst(pair(X, Y)) → a__U21(a__isLNat(X), X, Y)
a__head(cons(N, XS)) → a__U31(a__isNatural(N), N, XS)
a__isLNat(nil) → tt
a__isLNat(afterNth(V1, V2)) → a__U41(a__isNatural(V1), V2)
a__isLNat(cons(V1, V2)) → a__U51(a__isNatural(V1), V2)
a__isLNat(fst(V1)) → a__U61(a__isPLNat(V1))
a__isLNat(natsFrom(V1)) → a__U71(a__isNatural(V1))
a__isLNat(snd(V1)) → a__U81(a__isPLNat(V1))
a__isLNat(tail(V1)) → a__U91(a__isLNat(V1))
a__isLNat(take(V1, V2)) → a__U101(a__isNatural(V1), V2)
a__isNatural(0) → tt
a__isNatural(head(V1)) → a__U111(a__isLNat(V1))
a__isNatural(s(V1)) → a__U121(a__isNatural(V1))
a__isNatural(sel(V1, V2)) → a__U131(a__isNatural(V1), V2)
a__isPLNat(pair(V1, V2)) → a__U141(a__isLNat(V1), V2)
a__isPLNat(splitAt(V1, V2)) → a__U151(a__isNatural(V1), V2)
a__natsFrom(N) → a__U161(a__isNatural(N), N)
a__sel(N, XS) → a__U171(a__isNatural(N), N, XS)
a__snd(pair(X, Y)) → a__U181(a__isLNat(X), Y)
a__splitAt(0, XS) → a__U191(a__isLNat(XS), XS)
a__splitAt(s(N), cons(X, XS)) → a__U201(a__isNatural(N), N, X, XS)
a__tail(cons(N, XS)) → a__U211(a__isNatural(N), XS)
a__take(N, XS) → a__U221(a__isNatural(N), N, XS)
mark(U101(X1, X2)) → a__U101(mark(X1), X2)
mark(U102(X)) → a__U102(mark(X))
mark(isLNat(X)) → a__isLNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(U111(X)) → a__U111(mark(X))
mark(snd(X)) → a__snd(mark(X))
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(U121(X)) → a__U121(mark(X))
mark(U131(X1, X2)) → a__U131(mark(X1), X2)
mark(U132(X)) → a__U132(mark(X))
mark(U141(X1, X2)) → a__U141(mark(X1), X2)
mark(U142(X)) → a__U142(mark(X))
mark(U151(X1, X2)) → a__U151(mark(X1), X2)
mark(U152(X)) → a__U152(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(U172(X1, X2, X3)) → a__U172(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(U182(X1, X2)) → a__U182(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, X3, X4)) → a__U202(mark(X1), X2, X3, X4)
mark(isNatural(X)) → a__isNatural(X)
mark(U203(X1, X2, X3, X4)) → a__U203(mark(X1), X2, X3, X4)
mark(U204(X1, X2)) → a__U204(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U211(X1, X2)) → a__U211(mark(X1), X2)
mark(U212(X1, X2)) → a__U212(mark(X1), X2)
mark(U221(X1, X2, X3)) → a__U221(mark(X1), X2, X3)
mark(U222(X1, X2, X3)) → a__U222(mark(X1), X2, X3)
mark(fst(X)) → a__fst(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X)) → a__U91(mark(X))
mark(isPLNat(X)) → a__isPLNat(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) → U101(X1, X2)
a__U102(X) → U102(X)
a__isLNat(X) → isLNat(X)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U111(X) → U111(X)
a__snd(X) → snd(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U121(X) → U121(X)
a__U131(X1, X2) → U131(X1, X2)
a__U132(X) → U132(X)
a__U141(X1, X2) → U141(X1, X2)
a__U142(X) → U142(X)
a__U151(X1, X2) → U151(X1, X2)
a__U152(X) → U152(X)
a__U161(X1, X2) → U161(X1, X2)
a__natsFrom(X) → natsFrom(X)
a__U171(X1, X2, X3) → U171(X1, X2, X3)
a__U172(X1, X2, X3) → U172(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U181(X1, X2) → U181(X1, X2)
a__U182(X1, X2) → U182(X1, X2)
a__U191(X1, X2) → U191(X1, X2)
a__U201(X1, X2, X3, X4) → U201(X1, X2, X3, X4)
a__U202(X1, X2, X3, X4) → U202(X1, X2, X3, X4)
a__isNatural(X) → isNatural(X)
a__U203(X1, X2, X3, X4) → U203(X1, X2, X3, X4)
a__U204(X1, X2) → U204(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U211(X1, X2) → U211(X1, X2)
a__U212(X1, X2) → U212(X1, X2)
a__U221(X1, X2, X3) → U221(X1, X2, X3)
a__U222(X1, X2, X3) → U222(X1, X2, X3)
a__fst(X) → fst(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X) → U91(X)
a__isPLNat(X) → isPLNat(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, V2) → A__U102(a__isLNat(V2))
A__U101(tt, V2) → A__ISLNAT(V2)
A__U11(tt, N, XS) → A__U12(a__isLNat(XS), N, XS)
A__U11(tt, N, XS) → A__ISLNAT(XS)
A__U12(tt, N, XS) → A__SND(a__splitAt(mark(N), mark(XS)))
A__U12(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U12(tt, N, XS) → MARK(N)
A__U12(tt, N, XS) → MARK(XS)
A__U131(tt, V2) → A__U132(a__isLNat(V2))
A__U131(tt, V2) → A__ISLNAT(V2)
A__U141(tt, V2) → A__U142(a__isLNat(V2))
A__U141(tt, V2) → A__ISLNAT(V2)
A__U151(tt, V2) → A__U152(a__isLNat(V2))
A__U151(tt, V2) → A__ISLNAT(V2)
A__U161(tt, N) → MARK(N)
A__U171(tt, N, XS) → A__U172(a__isLNat(XS), N, XS)
A__U171(tt, N, XS) → A__ISLNAT(XS)
A__U172(tt, N, XS) → A__HEAD(a__afterNth(mark(N), mark(XS)))
A__U172(tt, N, XS) → A__AFTERNTH(mark(N), mark(XS))
A__U172(tt, N, XS) → MARK(N)
A__U172(tt, N, XS) → MARK(XS)
A__U181(tt, Y) → A__U182(a__isLNat(Y), Y)
A__U181(tt, Y) → A__ISLNAT(Y)
A__U182(tt, Y) → MARK(Y)
A__U191(tt, XS) → MARK(XS)
A__U201(tt, N, X, XS) → A__U202(a__isNatural(X), N, X, XS)
A__U201(tt, N, X, XS) → A__ISNATURAL(X)
A__U202(tt, N, X, XS) → A__U203(a__isLNat(XS), N, X, XS)
A__U202(tt, N, X, XS) → A__ISLNAT(XS)
A__U203(tt, N, X, XS) → A__U204(a__splitAt(mark(N), mark(XS)), X)
A__U203(tt, N, X, XS) → A__SPLITAT(mark(N), mark(XS))
A__U203(tt, N, X, XS) → MARK(N)
A__U203(tt, N, X, XS) → MARK(XS)
A__U204(pair(YS, ZS), X) → MARK(X)
A__U204(pair(YS, ZS), X) → MARK(ZS)
A__U21(tt, X, Y) → A__U22(a__isLNat(Y), X)
A__U21(tt, X, Y) → A__ISLNAT(Y)
A__U211(tt, XS) → A__U212(a__isLNat(XS), XS)
A__U211(tt, XS) → A__ISLNAT(XS)
A__U212(tt, XS) → MARK(XS)
A__U22(tt, X) → MARK(X)
A__U221(tt, N, XS) → A__U222(a__isLNat(XS), N, XS)
A__U221(tt, N, XS) → A__ISLNAT(XS)
A__U222(tt, N, XS) → A__FST(a__splitAt(mark(N), mark(XS)))
A__U222(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U222(tt, N, XS) → MARK(N)
A__U222(tt, N, XS) → MARK(XS)
A__U31(tt, N, XS) → A__U32(a__isLNat(XS), N)
A__U31(tt, N, XS) → A__ISLNAT(XS)
A__U32(tt, N) → MARK(N)
A__U41(tt, V2) → A__U42(a__isLNat(V2))
A__U41(tt, V2) → A__ISLNAT(V2)
A__U51(tt, V2) → A__U52(a__isLNat(V2))
A__U51(tt, V2) → A__ISLNAT(V2)
A__AFTERNTH(N, XS) → A__U11(a__isNatural(N), N, XS)
A__AFTERNTH(N, XS) → A__ISNATURAL(N)
A__FST(pair(X, Y)) → A__U21(a__isLNat(X), X, Y)
A__FST(pair(X, Y)) → A__ISLNAT(X)
A__HEAD(cons(N, XS)) → A__U31(a__isNatural(N), N, XS)
A__HEAD(cons(N, XS)) → A__ISNATURAL(N)
A__ISLNAT(afterNth(V1, V2)) → A__U41(a__isNatural(V1), V2)
A__ISLNAT(afterNth(V1, V2)) → A__ISNATURAL(V1)
A__ISLNAT(cons(V1, V2)) → A__U51(a__isNatural(V1), V2)
A__ISLNAT(cons(V1, V2)) → A__ISNATURAL(V1)
A__ISLNAT(fst(V1)) → A__U61(a__isPLNat(V1))
A__ISLNAT(fst(V1)) → A__ISPLNAT(V1)
A__ISLNAT(natsFrom(V1)) → A__U71(a__isNatural(V1))
A__ISLNAT(natsFrom(V1)) → A__ISNATURAL(V1)
A__ISLNAT(snd(V1)) → A__U81(a__isPLNat(V1))
A__ISLNAT(snd(V1)) → A__ISPLNAT(V1)
A__ISLNAT(tail(V1)) → A__U91(a__isLNat(V1))
A__ISLNAT(tail(V1)) → A__ISLNAT(V1)
A__ISLNAT(take(V1, V2)) → A__U101(a__isNatural(V1), V2)
A__ISLNAT(take(V1, V2)) → A__ISNATURAL(V1)
A__ISNATURAL(head(V1)) → A__U111(a__isLNat(V1))
A__ISNATURAL(head(V1)) → A__ISLNAT(V1)
A__ISNATURAL(s(V1)) → A__U121(a__isNatural(V1))
A__ISNATURAL(s(V1)) → A__ISNATURAL(V1)
A__ISNATURAL(sel(V1, V2)) → A__U131(a__isNatural(V1), V2)
A__ISNATURAL(sel(V1, V2)) → A__ISNATURAL(V1)
A__ISPLNAT(pair(V1, V2)) → A__U141(a__isLNat(V1), V2)
A__ISPLNAT(pair(V1, V2)) → A__ISLNAT(V1)
A__ISPLNAT(splitAt(V1, V2)) → A__U151(a__isNatural(V1), V2)
A__ISPLNAT(splitAt(V1, V2)) → A__ISNATURAL(V1)
A__NATSFROM(N) → A__U161(a__isNatural(N), N)
A__NATSFROM(N) → A__ISNATURAL(N)
A__SEL(N, XS) → A__U171(a__isNatural(N), N, XS)
A__SEL(N, XS) → A__ISNATURAL(N)
A__SND(pair(X, Y)) → A__U181(a__isLNat(X), Y)
A__SND(pair(X, Y)) → A__ISLNAT(X)
A__SPLITAT(0, XS) → A__U191(a__isLNat(XS), XS)
A__SPLITAT(0, XS) → A__ISLNAT(XS)
A__SPLITAT(s(N), cons(X, XS)) → A__U201(a__isNatural(N), N, X, XS)
A__SPLITAT(s(N), cons(X, XS)) → A__ISNATURAL(N)
A__TAIL(cons(N, XS)) → A__U211(a__isNatural(N), XS)
A__TAIL(cons(N, XS)) → A__ISNATURAL(N)
A__TAKE(N, XS) → A__U221(a__isNatural(N), N, XS)
A__TAKE(N, XS) → A__ISNATURAL(N)
MARK(U101(X1, X2)) → A__U101(mark(X1), X2)
MARK(U101(X1, X2)) → MARK(X1)
MARK(U102(X)) → A__U102(mark(X))
MARK(U102(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(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U111(X)) → A__U111(mark(X))
MARK(U111(X)) → MARK(X)
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(U121(X)) → A__U121(mark(X))
MARK(U121(X)) → MARK(X)
MARK(U131(X1, X2)) → A__U131(mark(X1), X2)
MARK(U131(X1, X2)) → MARK(X1)
MARK(U132(X)) → A__U132(mark(X))
MARK(U132(X)) → MARK(X)
MARK(U141(X1, X2)) → A__U141(mark(X1), X2)
MARK(U141(X1, X2)) → MARK(X1)
MARK(U142(X)) → A__U142(mark(X))
MARK(U142(X)) → MARK(X)
MARK(U151(X1, X2)) → A__U151(mark(X1), X2)
MARK(U151(X1, X2)) → MARK(X1)
MARK(U152(X)) → A__U152(mark(X))
MARK(U152(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(U172(X1, X2, X3)) → A__U172(mark(X1), X2, X3)
MARK(U172(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(U182(X1, X2)) → A__U182(mark(X1), X2)
MARK(U182(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, X3, X4)) → A__U202(mark(X1), X2, X3, X4)
MARK(U202(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatural(X)) → A__ISNATURAL(X)
MARK(U203(X1, X2, X3, X4)) → A__U203(mark(X1), X2, X3, X4)
MARK(U203(X1, X2, X3, X4)) → MARK(X1)
MARK(U204(X1, X2)) → A__U204(mark(X1), X2)
MARK(U204(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U211(X1, X2)) → A__U211(mark(X1), X2)
MARK(U211(X1, X2)) → MARK(X1)
MARK(U212(X1, X2)) → A__U212(mark(X1), X2)
MARK(U212(X1, X2)) → MARK(X1)
MARK(U221(X1, X2, X3)) → A__U221(mark(X1), X2, X3)
MARK(U221(X1, X2, X3)) → MARK(X1)
MARK(U222(X1, X2, X3)) → A__U222(mark(X1), X2, X3)
MARK(U222(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → A__FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → A__U42(mark(X))
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → A__U52(mark(X))
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → A__U61(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → A__U71(mark(X))
MARK(U71(X)) → MARK(X)
MARK(U81(X)) → A__U81(mark(X))
MARK(U81(X)) → MARK(X)
MARK(U91(X)) → A__U91(mark(X))
MARK(U91(X)) → MARK(X)
MARK(isPLNat(X)) → A__ISPLNAT(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, V2) → a__U102(a__isLNat(V2))
a__U102(tt) → tt
a__U11(tt, N, XS) → a__U12(a__isLNat(XS), N, XS)
a__U111(tt) → tt
a__U12(tt, N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__U121(tt) → tt
a__U131(tt, V2) → a__U132(a__isLNat(V2))
a__U132(tt) → tt
a__U141(tt, V2) → a__U142(a__isLNat(V2))
a__U142(tt) → tt
a__U151(tt, V2) → a__U152(a__isLNat(V2))
a__U152(tt) → tt
a__U161(tt, N) → cons(mark(N), natsFrom(s(N)))
a__U171(tt, N, XS) → a__U172(a__isLNat(XS), N, XS)
a__U172(tt, N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__U181(tt, Y) → a__U182(a__isLNat(Y), Y)
a__U182(tt, Y) → mark(Y)
a__U191(tt, XS) → pair(nil, mark(XS))
a__U201(tt, N, X, XS) → a__U202(a__isNatural(X), N, X, XS)
a__U202(tt, N, X, XS) → a__U203(a__isLNat(XS), N, X, XS)
a__U203(tt, N, X, XS) → a__U204(a__splitAt(mark(N), mark(XS)), X)
a__U204(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__U21(tt, X, Y) → a__U22(a__isLNat(Y), X)
a__U211(tt, XS) → a__U212(a__isLNat(XS), XS)
a__U212(tt, XS) → mark(XS)
a__U22(tt, X) → mark(X)
a__U221(tt, N, XS) → a__U222(a__isLNat(XS), N, XS)
a__U222(tt, N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
a__U31(tt, N, XS) → a__U32(a__isLNat(XS), N)
a__U32(tt, N) → mark(N)
a__U41(tt, V2) → a__U42(a__isLNat(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isLNat(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt) → tt
a__afterNth(N, XS) → a__U11(a__isNatural(N), N, XS)
a__fst(pair(X, Y)) → a__U21(a__isLNat(X), X, Y)
a__head(cons(N, XS)) → a__U31(a__isNatural(N), N, XS)
a__isLNat(nil) → tt
a__isLNat(afterNth(V1, V2)) → a__U41(a__isNatural(V1), V2)
a__isLNat(cons(V1, V2)) → a__U51(a__isNatural(V1), V2)
a__isLNat(fst(V1)) → a__U61(a__isPLNat(V1))
a__isLNat(natsFrom(V1)) → a__U71(a__isNatural(V1))
a__isLNat(snd(V1)) → a__U81(a__isPLNat(V1))
a__isLNat(tail(V1)) → a__U91(a__isLNat(V1))
a__isLNat(take(V1, V2)) → a__U101(a__isNatural(V1), V2)
a__isNatural(0) → tt
a__isNatural(head(V1)) → a__U111(a__isLNat(V1))
a__isNatural(s(V1)) → a__U121(a__isNatural(V1))
a__isNatural(sel(V1, V2)) → a__U131(a__isNatural(V1), V2)
a__isPLNat(pair(V1, V2)) → a__U141(a__isLNat(V1), V2)
a__isPLNat(splitAt(V1, V2)) → a__U151(a__isNatural(V1), V2)
a__natsFrom(N) → a__U161(a__isNatural(N), N)
a__sel(N, XS) → a__U171(a__isNatural(N), N, XS)
a__snd(pair(X, Y)) → a__U181(a__isLNat(X), Y)
a__splitAt(0, XS) → a__U191(a__isLNat(XS), XS)
a__splitAt(s(N), cons(X, XS)) → a__U201(a__isNatural(N), N, X, XS)
a__tail(cons(N, XS)) → a__U211(a__isNatural(N), XS)
a__take(N, XS) → a__U221(a__isNatural(N), N, XS)
mark(U101(X1, X2)) → a__U101(mark(X1), X2)
mark(U102(X)) → a__U102(mark(X))
mark(isLNat(X)) → a__isLNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(U111(X)) → a__U111(mark(X))
mark(snd(X)) → a__snd(mark(X))
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(U121(X)) → a__U121(mark(X))
mark(U131(X1, X2)) → a__U131(mark(X1), X2)
mark(U132(X)) → a__U132(mark(X))
mark(U141(X1, X2)) → a__U141(mark(X1), X2)
mark(U142(X)) → a__U142(mark(X))
mark(U151(X1, X2)) → a__U151(mark(X1), X2)
mark(U152(X)) → a__U152(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(U172(X1, X2, X3)) → a__U172(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(U182(X1, X2)) → a__U182(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, X3, X4)) → a__U202(mark(X1), X2, X3, X4)
mark(isNatural(X)) → a__isNatural(X)
mark(U203(X1, X2, X3, X4)) → a__U203(mark(X1), X2, X3, X4)
mark(U204(X1, X2)) → a__U204(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U211(X1, X2)) → a__U211(mark(X1), X2)
mark(U212(X1, X2)) → a__U212(mark(X1), X2)
mark(U221(X1, X2, X3)) → a__U221(mark(X1), X2, X3)
mark(U222(X1, X2, X3)) → a__U222(mark(X1), X2, X3)
mark(fst(X)) → a__fst(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X)) → a__U91(mark(X))
mark(isPLNat(X)) → a__isPLNat(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) → U101(X1, X2)
a__U102(X) → U102(X)
a__isLNat(X) → isLNat(X)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U111(X) → U111(X)
a__snd(X) → snd(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U121(X) → U121(X)
a__U131(X1, X2) → U131(X1, X2)
a__U132(X) → U132(X)
a__U141(X1, X2) → U141(X1, X2)
a__U142(X) → U142(X)
a__U151(X1, X2) → U151(X1, X2)
a__U152(X) → U152(X)
a__U161(X1, X2) → U161(X1, X2)
a__natsFrom(X) → natsFrom(X)
a__U171(X1, X2, X3) → U171(X1, X2, X3)
a__U172(X1, X2, X3) → U172(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U181(X1, X2) → U181(X1, X2)
a__U182(X1, X2) → U182(X1, X2)
a__U191(X1, X2) → U191(X1, X2)
a__U201(X1, X2, X3, X4) → U201(X1, X2, X3, X4)
a__U202(X1, X2, X3, X4) → U202(X1, X2, X3, X4)
a__isNatural(X) → isNatural(X)
a__U203(X1, X2, X3, X4) → U203(X1, X2, X3, X4)
a__U204(X1, X2) → U204(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U211(X1, X2) → U211(X1, X2)
a__U212(X1, X2) → U212(X1, X2)
a__U221(X1, X2, X3) → U221(X1, X2, X3)
a__U222(X1, X2, X3) → U222(X1, X2, X3)
a__fst(X) → fst(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X) → U91(X)
a__isPLNat(X) → isPLNat(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 2 SCCs with 52 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

A__U101(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(afterNth(V1, V2)) → A__U41(a__isNatural(V1), V2)
A__U41(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(afterNth(V1, V2)) → A__ISNATURAL(V1)
A__ISNATURAL(head(V1)) → A__ISLNAT(V1)
A__ISLNAT(cons(V1, V2)) → A__U51(a__isNatural(V1), V2)
A__U51(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(cons(V1, V2)) → A__ISNATURAL(V1)
A__ISNATURAL(s(V1)) → A__ISNATURAL(V1)
A__ISNATURAL(sel(V1, V2)) → A__U131(a__isNatural(V1), V2)
A__U131(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(fst(V1)) → A__ISPLNAT(V1)
A__ISPLNAT(pair(V1, V2)) → A__U141(a__isLNat(V1), V2)
A__U141(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(natsFrom(V1)) → A__ISNATURAL(V1)
A__ISNATURAL(sel(V1, V2)) → A__ISNATURAL(V1)
A__ISLNAT(snd(V1)) → A__ISPLNAT(V1)
A__ISPLNAT(pair(V1, V2)) → A__ISLNAT(V1)
A__ISLNAT(tail(V1)) → A__ISLNAT(V1)
A__ISLNAT(take(V1, V2)) → A__U101(a__isNatural(V1), V2)
A__ISLNAT(take(V1, V2)) → A__ISNATURAL(V1)
A__ISPLNAT(splitAt(V1, V2)) → A__U151(a__isNatural(V1), V2)
A__U151(tt, V2) → A__ISLNAT(V2)
A__ISPLNAT(splitAt(V1, V2)) → A__ISNATURAL(V1)

The TRS R consists of the following rules:

a__U101(tt, V2) → a__U102(a__isLNat(V2))
a__U102(tt) → tt
a__U11(tt, N, XS) → a__U12(a__isLNat(XS), N, XS)
a__U111(tt) → tt
a__U12(tt, N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__U121(tt) → tt
a__U131(tt, V2) → a__U132(a__isLNat(V2))
a__U132(tt) → tt
a__U141(tt, V2) → a__U142(a__isLNat(V2))
a__U142(tt) → tt
a__U151(tt, V2) → a__U152(a__isLNat(V2))
a__U152(tt) → tt
a__U161(tt, N) → cons(mark(N), natsFrom(s(N)))
a__U171(tt, N, XS) → a__U172(a__isLNat(XS), N, XS)
a__U172(tt, N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__U181(tt, Y) → a__U182(a__isLNat(Y), Y)
a__U182(tt, Y) → mark(Y)
a__U191(tt, XS) → pair(nil, mark(XS))
a__U201(tt, N, X, XS) → a__U202(a__isNatural(X), N, X, XS)
a__U202(tt, N, X, XS) → a__U203(a__isLNat(XS), N, X, XS)
a__U203(tt, N, X, XS) → a__U204(a__splitAt(mark(N), mark(XS)), X)
a__U204(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__U21(tt, X, Y) → a__U22(a__isLNat(Y), X)
a__U211(tt, XS) → a__U212(a__isLNat(XS), XS)
a__U212(tt, XS) → mark(XS)
a__U22(tt, X) → mark(X)
a__U221(tt, N, XS) → a__U222(a__isLNat(XS), N, XS)
a__U222(tt, N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
a__U31(tt, N, XS) → a__U32(a__isLNat(XS), N)
a__U32(tt, N) → mark(N)
a__U41(tt, V2) → a__U42(a__isLNat(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isLNat(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt) → tt
a__afterNth(N, XS) → a__U11(a__isNatural(N), N, XS)
a__fst(pair(X, Y)) → a__U21(a__isLNat(X), X, Y)
a__head(cons(N, XS)) → a__U31(a__isNatural(N), N, XS)
a__isLNat(nil) → tt
a__isLNat(afterNth(V1, V2)) → a__U41(a__isNatural(V1), V2)
a__isLNat(cons(V1, V2)) → a__U51(a__isNatural(V1), V2)
a__isLNat(fst(V1)) → a__U61(a__isPLNat(V1))
a__isLNat(natsFrom(V1)) → a__U71(a__isNatural(V1))
a__isLNat(snd(V1)) → a__U81(a__isPLNat(V1))
a__isLNat(tail(V1)) → a__U91(a__isLNat(V1))
a__isLNat(take(V1, V2)) → a__U101(a__isNatural(V1), V2)
a__isNatural(0) → tt
a__isNatural(head(V1)) → a__U111(a__isLNat(V1))
a__isNatural(s(V1)) → a__U121(a__isNatural(V1))
a__isNatural(sel(V1, V2)) → a__U131(a__isNatural(V1), V2)
a__isPLNat(pair(V1, V2)) → a__U141(a__isLNat(V1), V2)
a__isPLNat(splitAt(V1, V2)) → a__U151(a__isNatural(V1), V2)
a__natsFrom(N) → a__U161(a__isNatural(N), N)
a__sel(N, XS) → a__U171(a__isNatural(N), N, XS)
a__snd(pair(X, Y)) → a__U181(a__isLNat(X), Y)
a__splitAt(0, XS) → a__U191(a__isLNat(XS), XS)
a__splitAt(s(N), cons(X, XS)) → a__U201(a__isNatural(N), N, X, XS)
a__tail(cons(N, XS)) → a__U211(a__isNatural(N), XS)
a__take(N, XS) → a__U221(a__isNatural(N), N, XS)
mark(U101(X1, X2)) → a__U101(mark(X1), X2)
mark(U102(X)) → a__U102(mark(X))
mark(isLNat(X)) → a__isLNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(U111(X)) → a__U111(mark(X))
mark(snd(X)) → a__snd(mark(X))
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(U121(X)) → a__U121(mark(X))
mark(U131(X1, X2)) → a__U131(mark(X1), X2)
mark(U132(X)) → a__U132(mark(X))
mark(U141(X1, X2)) → a__U141(mark(X1), X2)
mark(U142(X)) → a__U142(mark(X))
mark(U151(X1, X2)) → a__U151(mark(X1), X2)
mark(U152(X)) → a__U152(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(U172(X1, X2, X3)) → a__U172(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(U182(X1, X2)) → a__U182(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, X3, X4)) → a__U202(mark(X1), X2, X3, X4)
mark(isNatural(X)) → a__isNatural(X)
mark(U203(X1, X2, X3, X4)) → a__U203(mark(X1), X2, X3, X4)
mark(U204(X1, X2)) → a__U204(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U211(X1, X2)) → a__U211(mark(X1), X2)
mark(U212(X1, X2)) → a__U212(mark(X1), X2)
mark(U221(X1, X2, X3)) → a__U221(mark(X1), X2, X3)
mark(U222(X1, X2, X3)) → a__U222(mark(X1), X2, X3)
mark(fst(X)) → a__fst(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X)) → a__U91(mark(X))
mark(isPLNat(X)) → a__isPLNat(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) → U101(X1, X2)
a__U102(X) → U102(X)
a__isLNat(X) → isLNat(X)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U111(X) → U111(X)
a__snd(X) → snd(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U121(X) → U121(X)
a__U131(X1, X2) → U131(X1, X2)
a__U132(X) → U132(X)
a__U141(X1, X2) → U141(X1, X2)
a__U142(X) → U142(X)
a__U151(X1, X2) → U151(X1, X2)
a__U152(X) → U152(X)
a__U161(X1, X2) → U161(X1, X2)
a__natsFrom(X) → natsFrom(X)
a__U171(X1, X2, X3) → U171(X1, X2, X3)
a__U172(X1, X2, X3) → U172(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U181(X1, X2) → U181(X1, X2)
a__U182(X1, X2) → U182(X1, X2)
a__U191(X1, X2) → U191(X1, X2)
a__U201(X1, X2, X3, X4) → U201(X1, X2, X3, X4)
a__U202(X1, X2, X3, X4) → U202(X1, X2, X3, X4)
a__isNatural(X) → isNatural(X)
a__U203(X1, X2, X3, X4) → U203(X1, X2, X3, X4)
a__U204(X1, X2) → U204(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U211(X1, X2) → U211(X1, X2)
a__U212(X1, X2) → U212(X1, X2)
a__U221(X1, X2, X3) → U221(X1, X2, X3)
a__U222(X1, X2, X3) → U222(X1, X2, X3)
a__fst(X) → fst(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X) → U91(X)
a__isPLNat(X) → isPLNat(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.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U101(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(afterNth(V1, V2)) → A__U41(a__isNatural(V1), V2)
A__ISLNAT(afterNth(V1, V2)) → A__ISNATURAL(V1)
A__ISNATURAL(head(V1)) → A__ISLNAT(V1)
A__ISLNAT(cons(V1, V2)) → A__U51(a__isNatural(V1), V2)
A__ISLNAT(cons(V1, V2)) → A__ISNATURAL(V1)
A__ISNATURAL(s(V1)) → A__ISNATURAL(V1)
A__ISNATURAL(sel(V1, V2)) → A__U131(a__isNatural(V1), V2)
A__U131(tt, V2) → A__ISLNAT(V2)
A__ISPLNAT(pair(V1, V2)) → A__U141(a__isLNat(V1), V2)
A__U141(tt, V2) → A__ISLNAT(V2)
A__ISNATURAL(sel(V1, V2)) → A__ISNATURAL(V1)
A__ISPLNAT(pair(V1, V2)) → A__ISLNAT(V1)
A__ISLNAT(tail(V1)) → A__ISLNAT(V1)
A__ISLNAT(take(V1, V2)) → A__U101(a__isNatural(V1), V2)
A__ISLNAT(take(V1, V2)) → A__ISNATURAL(V1)
A__ISPLNAT(splitAt(V1, V2)) → A__U151(a__isNatural(V1), V2)
A__U151(tt, V2) → A__ISLNAT(V2)
A__ISPLNAT(splitAt(V1, V2)) → A__ISNATURAL(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U101(x1, x2) = A__U101(x2)
tt = tt
A__ISLNAT(x1) = x1
afterNth(x1, x2) = afterNth(x1, x2)
A__U41(x1, x2) = x2
a__isNatural(x1) = a__isNatural
A__ISNATURAL(x1) = x1
head(x1) = head(x1)
cons(x1, x2) = cons(x1, x2)
A__U51(x1, x2) = x2
s(x1) = s(x1)
sel(x1, x2) = sel(x1, x2)
A__U131(x1, x2) = A__U131(x2)
fst(x1) = x1
A__ISPLNAT(x1) = x1
pair(x1, x2) = pair(x1, x2)
A__U141(x1, x2) = A__U141(x2)
a__isLNat(x1) = a__isLNat
natsFrom(x1) = x1
snd(x1) = x1
tail(x1) = tail(x1)
take(x1, x2) = take(x1, x2)
splitAt(x1, x2) = splitAt(x1, x2)
A__U151(x1, x2) = A__U151(x2)
isLNat(x1) = isLNat
a__U102(x1) = x1
U102(x1) = U102(x1)
a__U101(x1, x2) = a__U101
U101(x1, x2) = U101(x1, x2)
a__U61(x1) = x1
a__isPLNat(x1) = a__isPLNat
a__U51(x1, x2) = x2
a__U41(x1, x2) = x2
nil = nil
a__U91(x1) = x1
a__U81(x1) = a__U81(x1)
a__U71(x1) = x1
a__U131(x1, x2) = a__U131(x2)
a__U121(x1) = x1
a__U111(x1) = x1
0 = 0
a__U151(x1, x2) = x2
a__U141(x1, x2) = a__U141
isPLNat(x1) = isPLNat(x1)
U91(x1) = U91(x1)
U81(x1) = U81(x1)
U71(x1) = U71(x1)
a__U42(x1) = x1
a__U52(x1) = x1
U41(x1, x2) = U41(x1, x2)
U61(x1) = U61(x1)
U52(x1) = U52(x1)
U51(x1, x2) = U51(x1, x2)
U42(x1) = U42(x1)
a__U152(x1) = x1
a__U142(x1) = x1
isNatural(x1) = isNatural(x1)
U142(x1) = U142(x1)
U141(x1, x2) = U141(x2)
U152(x1) = U152
U151(x1, x2) = U151
U111(x1) = U111
U121(x1) = x1
a__U132(x1) = x1
U132(x1) = U132(x1)
U131(x1, x2) = U131(x1, x2)

Recursive Path Order [RPO].
Precedence:

afterNth₂ > a_{_isPLNat}
head₁ > a_{_isPLNat}
cons₂ > a_{_isPLNat}
s₁ > a_{_isPLNat}
sel₂ > A_{_U131₁} > a_{_isPLNat}
pair₂ > A_{_U141₁} > a_{_isPLNat}
a_{_isLNat} > tt > a_{_isPLNat}
a_{_isLNat} > a_{_isNatural} > isNatural₁ > a_{_isPLNat}
a_{_isLNat} > isLNat > a_{_isPLNat}
a_{_isLNat} > a_{_U101} > a_{_isPLNat}
a_{_isLNat} > a_{_U81₁} > a_{_isPLNat}
tail₁ > a_{_isPLNat}
take₂ > A_{_U101₁} > a_{_isPLNat}
take₂ > a_{_isNatural} > isNatural₁ > a_{_isPLNat}
take₂ > a_{_U101} > a_{_isPLNat}
splitAt₂ > A_{_U151₁} > a_{_isPLNat}
U102₁ > a_{_isPLNat}
U101₂ > a_{_isPLNat}
nil > a_{_isPLNat}
a_{_U131₁} > U131₂ > a_{_isPLNat}
0 > tt > a_{_isPLNat}
a_{_U141} > U141₁ > a_{_isPLNat}
isPLNat₁ > a_{_isPLNat}
U91₁ > a_{_isPLNat}
U81₁ > a_{_isPLNat}
U71₁ > a_{_isPLNat}
U41₂ > a_{_isPLNat}
U61₁ > a_{_isPLNat}
U52₁ > a_{_isPLNat}
U51₂ > a_{_isPLNat}
U42₁ > a_{_isPLNat}
U142₁ > a_{_isPLNat}
U152 > a_{_isPLNat}
U151 > a_{_isPLNat}
U111 > a_{_isPLNat}
U132₁ > a_{_isPLNat}

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

A__U41(tt, V2) → A__ISLNAT(V2)
A__U51(tt, V2) → A__ISLNAT(V2)
A__ISLNAT(fst(V1)) → A__ISPLNAT(V1)
A__ISLNAT(natsFrom(V1)) → A__ISNATURAL(V1)
A__ISLNAT(snd(V1)) → A__ISPLNAT(V1)

The TRS R consists of the following rules:

a__U101(tt, V2) → a__U102(a__isLNat(V2))
a__U102(tt) → tt
a__U11(tt, N, XS) → a__U12(a__isLNat(XS), N, XS)
a__U111(tt) → tt
a__U12(tt, N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__U121(tt) → tt
a__U131(tt, V2) → a__U132(a__isLNat(V2))
a__U132(tt) → tt
a__U141(tt, V2) → a__U142(a__isLNat(V2))
a__U142(tt) → tt
a__U151(tt, V2) → a__U152(a__isLNat(V2))
a__U152(tt) → tt
a__U161(tt, N) → cons(mark(N), natsFrom(s(N)))
a__U171(tt, N, XS) → a__U172(a__isLNat(XS), N, XS)
a__U172(tt, N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__U181(tt, Y) → a__U182(a__isLNat(Y), Y)
a__U182(tt, Y) → mark(Y)
a__U191(tt, XS) → pair(nil, mark(XS))
a__U201(tt, N, X, XS) → a__U202(a__isNatural(X), N, X, XS)
a__U202(tt, N, X, XS) → a__U203(a__isLNat(XS), N, X, XS)
a__U203(tt, N, X, XS) → a__U204(a__splitAt(mark(N), mark(XS)), X)
a__U204(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__U21(tt, X, Y) → a__U22(a__isLNat(Y), X)
a__U211(tt, XS) → a__U212(a__isLNat(XS), XS)
a__U212(tt, XS) → mark(XS)
a__U22(tt, X) → mark(X)
a__U221(tt, N, XS) → a__U222(a__isLNat(XS), N, XS)
a__U222(tt, N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
a__U31(tt, N, XS) → a__U32(a__isLNat(XS), N)
a__U32(tt, N) → mark(N)
a__U41(tt, V2) → a__U42(a__isLNat(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isLNat(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt) → tt
a__afterNth(N, XS) → a__U11(a__isNatural(N), N, XS)
a__fst(pair(X, Y)) → a__U21(a__isLNat(X), X, Y)
a__head(cons(N, XS)) → a__U31(a__isNatural(N), N, XS)
a__isLNat(nil) → tt
a__isLNat(afterNth(V1, V2)) → a__U41(a__isNatural(V1), V2)
a__isLNat(cons(V1, V2)) → a__U51(a__isNatural(V1), V2)
a__isLNat(fst(V1)) → a__U61(a__isPLNat(V1))
a__isLNat(natsFrom(V1)) → a__U71(a__isNatural(V1))
a__isLNat(snd(V1)) → a__U81(a__isPLNat(V1))
a__isLNat(tail(V1)) → a__U91(a__isLNat(V1))
a__isLNat(take(V1, V2)) → a__U101(a__isNatural(V1), V2)
a__isNatural(0) → tt
a__isNatural(head(V1)) → a__U111(a__isLNat(V1))
a__isNatural(s(V1)) → a__U121(a__isNatural(V1))
a__isNatural(sel(V1, V2)) → a__U131(a__isNatural(V1), V2)
a__isPLNat(pair(V1, V2)) → a__U141(a__isLNat(V1), V2)
a__isPLNat(splitAt(V1, V2)) → a__U151(a__isNatural(V1), V2)
a__natsFrom(N) → a__U161(a__isNatural(N), N)
a__sel(N, XS) → a__U171(a__isNatural(N), N, XS)
a__snd(pair(X, Y)) → a__U181(a__isLNat(X), Y)
a__splitAt(0, XS) → a__U191(a__isLNat(XS), XS)
a__splitAt(s(N), cons(X, XS)) → a__U201(a__isNatural(N), N, X, XS)
a__tail(cons(N, XS)) → a__U211(a__isNatural(N), XS)
a__take(N, XS) → a__U221(a__isNatural(N), N, XS)
mark(U101(X1, X2)) → a__U101(mark(X1), X2)
mark(U102(X)) → a__U102(mark(X))
mark(isLNat(X)) → a__isLNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(U111(X)) → a__U111(mark(X))
mark(snd(X)) → a__snd(mark(X))
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(U121(X)) → a__U121(mark(X))
mark(U131(X1, X2)) → a__U131(mark(X1), X2)
mark(U132(X)) → a__U132(mark(X))
mark(U141(X1, X2)) → a__U141(mark(X1), X2)
mark(U142(X)) → a__U142(mark(X))
mark(U151(X1, X2)) → a__U151(mark(X1), X2)
mark(U152(X)) → a__U152(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(U172(X1, X2, X3)) → a__U172(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(U182(X1, X2)) → a__U182(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, X3, X4)) → a__U202(mark(X1), X2, X3, X4)
mark(isNatural(X)) → a__isNatural(X)
mark(U203(X1, X2, X3, X4)) → a__U203(mark(X1), X2, X3, X4)
mark(U204(X1, X2)) → a__U204(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U211(X1, X2)) → a__U211(mark(X1), X2)
mark(U212(X1, X2)) → a__U212(mark(X1), X2)
mark(U221(X1, X2, X3)) → a__U221(mark(X1), X2, X3)
mark(U222(X1, X2, X3)) → a__U222(mark(X1), X2, X3)
mark(fst(X)) → a__fst(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X)) → a__U91(mark(X))
mark(isPLNat(X)) → a__isPLNat(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) → U101(X1, X2)
a__U102(X) → U102(X)
a__isLNat(X) → isLNat(X)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U111(X) → U111(X)
a__snd(X) → snd(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U121(X) → U121(X)
a__U131(X1, X2) → U131(X1, X2)
a__U132(X) → U132(X)
a__U141(X1, X2) → U141(X1, X2)
a__U142(X) → U142(X)
a__U151(X1, X2) → U151(X1, X2)
a__U152(X) → U152(X)
a__U161(X1, X2) → U161(X1, X2)
a__natsFrom(X) → natsFrom(X)
a__U171(X1, X2, X3) → U171(X1, X2, X3)
a__U172(X1, X2, X3) → U172(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U181(X1, X2) → U181(X1, X2)
a__U182(X1, X2) → U182(X1, X2)
a__U191(X1, X2) → U191(X1, X2)
a__U201(X1, X2, X3, X4) → U201(X1, X2, X3, X4)
a__U202(X1, X2, X3, X4) → U202(X1, X2, X3, X4)
a__isNatural(X) → isNatural(X)
a__U203(X1, X2, X3, X4) → U203(X1, X2, X3, X4)
a__U204(X1, X2) → U204(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U211(X1, X2) → U211(X1, X2)
a__U212(X1, X2) → U212(X1, X2)
a__U221(X1, X2, X3) → U221(X1, X2, X3)
a__U222(X1, X2, X3) → U222(X1, X2, X3)
a__fst(X) → fst(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X) → U91(X)
a__isPLNat(X) → isPLNat(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.

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.

(9) TRUE

(10) Obligation:

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

MARK(U101(X1, X2)) → MARK(X1)
MARK(U102(X)) → MARK(X)
MARK(U11(X1, X2, X3)) → A__U11(mark(X1), X2, X3)
A__U11(tt, N, XS) → A__U12(a__isLNat(XS), N, XS)
A__U12(tt, N, XS) → A__SND(a__splitAt(mark(N), mark(XS)))
A__SND(pair(X, Y)) → A__U181(a__isLNat(X), Y)
A__U181(tt, Y) → A__U182(a__isLNat(Y), Y)
A__U182(tt, Y) → MARK(Y)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → A__U12(mark(X1), X2, X3)
A__U12(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__SPLITAT(0, XS) → A__U191(a__isLNat(XS), XS)
A__U191(tt, XS) → MARK(XS)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U111(X)) → MARK(X)
MARK(snd(X)) → A__SND(mark(X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → A__SPLITAT(mark(X1), mark(X2))
A__SPLITAT(s(N), cons(X, XS)) → A__U201(a__isNatural(N), N, X, XS)
A__U201(tt, N, X, XS) → A__U202(a__isNatural(X), N, X, XS)
A__U202(tt, N, X, XS) → A__U203(a__isLNat(XS), N, X, XS)
A__U203(tt, N, X, XS) → A__U204(a__splitAt(mark(N), mark(XS)), X)
A__U204(pair(YS, ZS), X) → MARK(X)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U121(X)) → MARK(X)
MARK(U131(X1, X2)) → MARK(X1)
MARK(U132(X)) → MARK(X)
MARK(U141(X1, X2)) → MARK(X1)
MARK(U142(X)) → MARK(X)
MARK(U151(X1, X2)) → MARK(X1)
MARK(U152(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__isNatural(N), N)
MARK(natsFrom(X)) → MARK(X)
MARK(U171(X1, X2, X3)) → A__U171(mark(X1), X2, X3)
A__U171(tt, N, XS) → A__U172(a__isLNat(XS), N, XS)
A__U172(tt, N, XS) → A__HEAD(a__afterNth(mark(N), mark(XS)))
A__HEAD(cons(N, XS)) → A__U31(a__isNatural(N), N, XS)
A__U31(tt, N, XS) → A__U32(a__isLNat(XS), N)
A__U32(tt, N) → MARK(N)
MARK(U171(X1, X2, X3)) → MARK(X1)
MARK(U172(X1, X2, X3)) → A__U172(mark(X1), X2, X3)
A__U172(tt, N, XS) → A__AFTERNTH(mark(N), mark(XS))
A__AFTERNTH(N, XS) → A__U11(a__isNatural(N), N, XS)
A__U172(tt, N, XS) → MARK(N)
MARK(U172(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(U182(X1, X2)) → A__U182(mark(X1), X2)
MARK(U182(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, X3, X4)) → A__U202(mark(X1), X2, X3, X4)
MARK(U202(X1, X2, X3, X4)) → MARK(X1)
MARK(U203(X1, X2, X3, X4)) → A__U203(mark(X1), X2, X3, X4)
A__U203(tt, N, X, XS) → A__SPLITAT(mark(N), mark(XS))
A__U203(tt, N, X, XS) → MARK(N)
MARK(U203(X1, X2, X3, X4)) → MARK(X1)
MARK(U204(X1, X2)) → A__U204(mark(X1), X2)
A__U204(pair(YS, ZS), X) → MARK(ZS)
MARK(U204(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
A__U21(tt, X, Y) → A__U22(a__isLNat(Y), X)
A__U22(tt, X) → MARK(X)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U211(X1, X2)) → A__U211(mark(X1), X2)
A__U211(tt, XS) → A__U212(a__isLNat(XS), XS)
A__U212(tt, XS) → MARK(XS)
MARK(U211(X1, X2)) → MARK(X1)
MARK(U212(X1, X2)) → A__U212(mark(X1), X2)
MARK(U212(X1, X2)) → MARK(X1)
MARK(U221(X1, X2, X3)) → A__U221(mark(X1), X2, X3)
A__U221(tt, N, XS) → A__U222(a__isLNat(XS), N, XS)
A__U222(tt, N, XS) → A__FST(a__splitAt(mark(N), mark(XS)))
A__FST(pair(X, Y)) → A__U21(a__isLNat(X), X, Y)
A__U222(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U222(tt, N, XS) → MARK(N)
MARK(U221(X1, X2, X3)) → MARK(X1)
MARK(U222(X1, X2, X3)) → A__U222(mark(X1), X2, X3)
A__U222(tt, N, XS) → MARK(XS)
MARK(U222(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → A__FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → A__U31(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
MARK(U91(X)) → MARK(X)
MARK(tail(X)) → A__TAIL(mark(X))
A__TAIL(cons(N, XS)) → A__U211(a__isNatural(N), XS)
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(N, XS) → A__U221(a__isNatural(N), N, 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__isNatural(N), N, XS)
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__U203(tt, N, X, XS) → MARK(XS)
A__U172(tt, N, XS) → MARK(XS)
A__U12(tt, N, XS) → MARK(N)
A__U12(tt, N, XS) → MARK(XS)

The TRS R consists of the following rules:

a__U101(tt, V2) → a__U102(a__isLNat(V2))
a__U102(tt) → tt
a__U11(tt, N, XS) → a__U12(a__isLNat(XS), N, XS)
a__U111(tt) → tt
a__U12(tt, N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__U121(tt) → tt
a__U131(tt, V2) → a__U132(a__isLNat(V2))
a__U132(tt) → tt
a__U141(tt, V2) → a__U142(a__isLNat(V2))
a__U142(tt) → tt
a__U151(tt, V2) → a__U152(a__isLNat(V2))
a__U152(tt) → tt
a__U161(tt, N) → cons(mark(N), natsFrom(s(N)))
a__U171(tt, N, XS) → a__U172(a__isLNat(XS), N, XS)
a__U172(tt, N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__U181(tt, Y) → a__U182(a__isLNat(Y), Y)
a__U182(tt, Y) → mark(Y)
a__U191(tt, XS) → pair(nil, mark(XS))
a__U201(tt, N, X, XS) → a__U202(a__isNatural(X), N, X, XS)
a__U202(tt, N, X, XS) → a__U203(a__isLNat(XS), N, X, XS)
a__U203(tt, N, X, XS) → a__U204(a__splitAt(mark(N), mark(XS)), X)
a__U204(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__U21(tt, X, Y) → a__U22(a__isLNat(Y), X)
a__U211(tt, XS) → a__U212(a__isLNat(XS), XS)
a__U212(tt, XS) → mark(XS)
a__U22(tt, X) → mark(X)
a__U221(tt, N, XS) → a__U222(a__isLNat(XS), N, XS)
a__U222(tt, N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
a__U31(tt, N, XS) → a__U32(a__isLNat(XS), N)
a__U32(tt, N) → mark(N)
a__U41(tt, V2) → a__U42(a__isLNat(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isLNat(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt) → tt
a__afterNth(N, XS) → a__U11(a__isNatural(N), N, XS)
a__fst(pair(X, Y)) → a__U21(a__isLNat(X), X, Y)
a__head(cons(N, XS)) → a__U31(a__isNatural(N), N, XS)
a__isLNat(nil) → tt
a__isLNat(afterNth(V1, V2)) → a__U41(a__isNatural(V1), V2)
a__isLNat(cons(V1, V2)) → a__U51(a__isNatural(V1), V2)
a__isLNat(fst(V1)) → a__U61(a__isPLNat(V1))
a__isLNat(natsFrom(V1)) → a__U71(a__isNatural(V1))
a__isLNat(snd(V1)) → a__U81(a__isPLNat(V1))
a__isLNat(tail(V1)) → a__U91(a__isLNat(V1))
a__isLNat(take(V1, V2)) → a__U101(a__isNatural(V1), V2)
a__isNatural(0) → tt
a__isNatural(head(V1)) → a__U111(a__isLNat(V1))
a__isNatural(s(V1)) → a__U121(a__isNatural(V1))
a__isNatural(sel(V1, V2)) → a__U131(a__isNatural(V1), V2)
a__isPLNat(pair(V1, V2)) → a__U141(a__isLNat(V1), V2)
a__isPLNat(splitAt(V1, V2)) → a__U151(a__isNatural(V1), V2)
a__natsFrom(N) → a__U161(a__isNatural(N), N)
a__sel(N, XS) → a__U171(a__isNatural(N), N, XS)
a__snd(pair(X, Y)) → a__U181(a__isLNat(X), Y)
a__splitAt(0, XS) → a__U191(a__isLNat(XS), XS)
a__splitAt(s(N), cons(X, XS)) → a__U201(a__isNatural(N), N, X, XS)
a__tail(cons(N, XS)) → a__U211(a__isNatural(N), XS)
a__take(N, XS) → a__U221(a__isNatural(N), N, XS)
mark(U101(X1, X2)) → a__U101(mark(X1), X2)
mark(U102(X)) → a__U102(mark(X))
mark(isLNat(X)) → a__isLNat(X)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(U111(X)) → a__U111(mark(X))
mark(snd(X)) → a__snd(mark(X))
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(U121(X)) → a__U121(mark(X))
mark(U131(X1, X2)) → a__U131(mark(X1), X2)
mark(U132(X)) → a__U132(mark(X))
mark(U141(X1, X2)) → a__U141(mark(X1), X2)
mark(U142(X)) → a__U142(mark(X))
mark(U151(X1, X2)) → a__U151(mark(X1), X2)
mark(U152(X)) → a__U152(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(U172(X1, X2, X3)) → a__U172(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(U182(X1, X2)) → a__U182(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, X3, X4)) → a__U202(mark(X1), X2, X3, X4)
mark(isNatural(X)) → a__isNatural(X)
mark(U203(X1, X2, X3, X4)) → a__U203(mark(X1), X2, X3, X4)
mark(U204(X1, X2)) → a__U204(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U211(X1, X2)) → a__U211(mark(X1), X2)
mark(U212(X1, X2)) → a__U212(mark(X1), X2)
mark(U221(X1, X2, X3)) → a__U221(mark(X1), X2, X3)
mark(U222(X1, X2, X3)) → a__U222(mark(X1), X2, X3)
mark(fst(X)) → a__fst(mark(X))
mark(U31(X1, X2, X3)) → a__U31(mark(X1), X2, X3)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X)) → a__U91(mark(X))
mark(isPLNat(X)) → a__isPLNat(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) → U101(X1, X2)
a__U102(X) → U102(X)
a__isLNat(X) → isLNat(X)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__U111(X) → U111(X)
a__snd(X) → snd(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U121(X) → U121(X)
a__U131(X1, X2) → U131(X1, X2)
a__U132(X) → U132(X)
a__U141(X1, X2) → U141(X1, X2)
a__U142(X) → U142(X)
a__U151(X1, X2) → U151(X1, X2)
a__U152(X) → U152(X)
a__U161(X1, X2) → U161(X1, X2)
a__natsFrom(X) → natsFrom(X)
a__U171(X1, X2, X3) → U171(X1, X2, X3)
a__U172(X1, X2, X3) → U172(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U181(X1, X2) → U181(X1, X2)
a__U182(X1, X2) → U182(X1, X2)
a__U191(X1, X2) → U191(X1, X2)
a__U201(X1, X2, X3, X4) → U201(X1, X2, X3, X4)
a__U202(X1, X2, X3, X4) → U202(X1, X2, X3, X4)
a__isNatural(X) → isNatural(X)
a__U203(X1, X2, X3, X4) → U203(X1, X2, X3, X4)
a__U204(X1, X2) → U204(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U211(X1, X2) → U211(X1, X2)
a__U212(X1, X2) → U212(X1, X2)
a__U221(X1, X2, X3) → U221(X1, X2, X3)
a__U222(X1, X2, X3) → U222(X1, X2, X3)
a__fst(X) → fst(X)
a__U31(X1, X2, X3) → U31(X1, X2, X3)
a__U32(X1, X2) → U32(X1, X2)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X) → U91(X)
a__isPLNat(X) → isPLNat(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.