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__U11(tt, N, XS) → a__U12(tt, N, XS)
a__U12(tt, N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__U21(tt, X) → a__U22(tt, X)
a__U22(tt, X) → mark(X)
a__U31(tt, N) → a__U32(tt, N)
a__U32(tt, N) → mark(N)
a__U41(tt, N, XS) → a__U42(tt, N, XS)
a__U42(tt, N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__U51(tt, Y) → a__U52(tt, Y)
a__U52(tt, Y) → mark(Y)
a__U61(tt, N, X, XS) → a__U62(tt, N, X, XS)
a__U62(tt, N, X, XS) → a__U63(tt, N, X, XS)
a__U63(tt, N, X, XS) → a__U64(a__splitAt(mark(N), mark(XS)), X)
a__U64(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__U71(tt, XS) → a__U72(tt, XS)
a__U72(tt, XS) → mark(XS)
a__U81(tt, N, XS) → a__U82(tt, N, XS)
a__U82(tt, N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
a__afterNth(N, XS) → a__U11(tt, N, XS)
a__fst(pair(X, Y)) → a__U21(tt, X)
a__head(cons(N, XS)) → a__U31(tt, N)
a__natsFrom(N) → cons(mark(N), natsFrom(s(N)))
a__sel(N, XS) → a__U41(tt, N, XS)
a__snd(pair(X, Y)) → a__U51(tt, Y)
a__splitAt(0, XS) → pair(nil, mark(XS))
a__splitAt(s(N), cons(X, XS)) → a__U61(tt, N, X, XS)
a__tail(cons(N, XS)) → a__U71(tt, XS)
a__take(N, XS) → a__U81(tt, N, XS)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(snd(X)) → a__snd(mark(X))
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(head(X)) → a__head(mark(X))
mark(afterNth(X1, X2)) → a__afterNth(mark(X1), mark(X2))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3, X4)) → a__U61(mark(X1), X2, X3, X4)
mark(U62(X1, X2, X3, X4)) → a__U62(mark(X1), X2, X3, X4)
mark(U63(X1, X2, X3, X4)) → a__U63(mark(X1), X2, X3, X4)
mark(U64(X1, X2)) → a__U64(mark(X1), X2)
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(fst(X)) → a__fst(mark(X))
mark(natsFrom(X)) → a__natsFrom(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(tt) → tt
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__snd(X) → snd(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3, X4) → U61(X1, X2, X3, X4)
a__U62(X1, X2, X3, X4) → U62(X1, X2, X3, X4)
a__U63(X1, X2, X3, X4) → U63(X1, X2, X3, X4)
a__U64(X1, X2) → U64(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__fst(X) → fst(X)
a__natsFrom(X) → natsFrom(X)
a__sel(X1, X2) → sel(X1, X2)
a__tail(X) → tail(X)
a__take(X1, X2) → take(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__U11(tt, N, XS) → A__U12(tt, N, 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__U21(tt, X) → A__U22(tt, X)
A__U22(tt, X) → MARK(X)
A__U31(tt, N) → A__U32(tt, N)
A__U32(tt, N) → MARK(N)
A__U41(tt, N, XS) → A__U42(tt, N, XS)
A__U42(tt, N, XS) → A__HEAD(a__afterNth(mark(N), mark(XS)))
A__U42(tt, N, XS) → A__AFTERNTH(mark(N), mark(XS))
A__U42(tt, N, XS) → MARK(N)
A__U42(tt, N, XS) → MARK(XS)
A__U51(tt, Y) → A__U52(tt, Y)
A__U52(tt, Y) → MARK(Y)
A__U61(tt, N, X, XS) → A__U62(tt, N, X, XS)
A__U62(tt, N, X, XS) → A__U63(tt, N, X, XS)
A__U63(tt, N, X, XS) → A__U64(a__splitAt(mark(N), mark(XS)), X)
A__U63(tt, N, X, XS) → A__SPLITAT(mark(N), mark(XS))
A__U63(tt, N, X, XS) → MARK(N)
A__U63(tt, N, X, XS) → MARK(XS)
A__U64(pair(YS, ZS), X) → MARK(X)
A__U64(pair(YS, ZS), X) → MARK(ZS)
A__U71(tt, XS) → A__U72(tt, XS)
A__U72(tt, XS) → MARK(XS)
A__U81(tt, N, XS) → A__U82(tt, N, XS)
A__U82(tt, N, XS) → A__FST(a__splitAt(mark(N), mark(XS)))
A__U82(tt, N, XS) → A__SPLITAT(mark(N), mark(XS))
A__U82(tt, N, XS) → MARK(N)
A__U82(tt, N, XS) → MARK(XS)
A__AFTERNTH(N, XS) → A__U11(tt, N, XS)
A__FST(pair(X, Y)) → A__U21(tt, X)
A__HEAD(cons(N, XS)) → A__U31(tt, N)
A__NATSFROM(N) → MARK(N)
A__SEL(N, XS) → A__U41(tt, N, XS)
A__SND(pair(X, Y)) → A__U51(tt, Y)
A__SPLITAT(0, XS) → MARK(XS)
A__SPLITAT(s(N), cons(X, XS)) → A__U61(tt, N, X, XS)
A__TAIL(cons(N, XS)) → A__U71(tt, XS)
A__TAKE(N, XS) → A__U81(tt, N, XS)
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(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(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(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, X3)) → A__U42(mark(X1), X2, X3)
MARK(U42(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(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3, X4)) → A__U61(mark(X1), X2, X3, X4)
MARK(U61(X1, X2, X3, X4)) → MARK(X1)
MARK(U62(X1, X2, X3, X4)) → A__U62(mark(X1), X2, X3, X4)
MARK(U62(X1, X2, X3, X4)) → MARK(X1)
MARK(U63(X1, X2, X3, X4)) → A__U63(mark(X1), X2, X3, X4)
MARK(U63(X1, X2, X3, X4)) → MARK(X1)
MARK(U64(X1, X2)) → A__U64(mark(X1), X2)
MARK(U64(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → A__U81(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → A__U82(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → A__FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(natsFrom(X)) → A__NATSFROM(mark(X))
MARK(natsFrom(X)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
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(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__U11(tt, N, XS) → a__U12(tt, N, XS)
a__U12(tt, N, XS) → a__snd(a__splitAt(mark(N), mark(XS)))
a__U21(tt, X) → a__U22(tt, X)
a__U22(tt, X) → mark(X)
a__U31(tt, N) → a__U32(tt, N)
a__U32(tt, N) → mark(N)
a__U41(tt, N, XS) → a__U42(tt, N, XS)
a__U42(tt, N, XS) → a__head(a__afterNth(mark(N), mark(XS)))
a__U51(tt, Y) → a__U52(tt, Y)
a__U52(tt, Y) → mark(Y)
a__U61(tt, N, X, XS) → a__U62(tt, N, X, XS)
a__U62(tt, N, X, XS) → a__U63(tt, N, X, XS)
a__U63(tt, N, X, XS) → a__U64(a__splitAt(mark(N), mark(XS)), X)
a__U64(pair(YS, ZS), X) → pair(cons(mark(X), YS), mark(ZS))
a__U71(tt, XS) → a__U72(tt, XS)
a__U72(tt, XS) → mark(XS)
a__U81(tt, N, XS) → a__U82(tt, N, XS)
a__U82(tt, N, XS) → a__fst(a__splitAt(mark(N), mark(XS)))
a__afterNth(N, XS) → a__U11(tt, N, XS)
a__fst(pair(X, Y)) → a__U21(tt, X)
a__head(cons(N, XS)) → a__U31(tt, N)
a__natsFrom(N) → cons(mark(N), natsFrom(s(N)))
a__sel(N, XS) → a__U41(tt, N, XS)
a__snd(pair(X, Y)) → a__U51(tt, Y)
a__splitAt(0, XS) → pair(nil, mark(XS))
a__splitAt(s(N), cons(X, XS)) → a__U61(tt, N, X, XS)
a__tail(cons(N, XS)) → a__U71(tt, XS)
a__take(N, XS) → a__U81(tt, N, XS)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(snd(X)) → a__snd(mark(X))
mark(splitAt(X1, X2)) → a__splitAt(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(head(X)) → a__head(mark(X))
mark(afterNth(X1, X2)) → a__afterNth(mark(X1), mark(X2))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U61(X1, X2, X3, X4)) → a__U61(mark(X1), X2, X3, X4)
mark(U62(X1, X2, X3, X4)) → a__U62(mark(X1), X2, X3, X4)
mark(U63(X1, X2, X3, X4)) → a__U63(mark(X1), X2, X3, X4)
mark(U64(X1, X2)) → a__U64(mark(X1), X2)
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U81(X1, X2, X3)) → a__U81(mark(X1), X2, X3)
mark(U82(X1, X2, X3)) → a__U82(mark(X1), X2, X3)
mark(fst(X)) → a__fst(mark(X))
mark(natsFrom(X)) → a__natsFrom(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(tail(X)) → a__tail(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(tt) → tt
mark(pair(X1, X2)) → pair(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__snd(X) → snd(X)
a__splitAt(X1, X2) → splitAt(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__head(X) → head(X)
a__afterNth(X1, X2) → afterNth(X1, X2)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X1, X2) → U52(X1, X2)
a__U61(X1, X2, X3, X4) → U61(X1, X2, X3, X4)
a__U62(X1, X2, X3, X4) → U62(X1, X2, X3, X4)
a__U63(X1, X2, X3, X4) → U63(X1, X2, X3, X4)
a__U64(X1, X2) → U64(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X1, X2) → U72(X1, X2)
a__U81(X1, X2, X3) → U81(X1, X2, X3)
a__U82(X1, X2, X3) → U82(X1, X2, X3)
a__fst(X) → fst(X)
a__natsFrom(X) → natsFrom(X)
a__sel(X1, X2) → sel(X1, X2)
a__tail(X) → tail(X)
a__take(X1, X2) → take(X1, X2)

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