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:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(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:

ACTIVE(U11(tt, N, XS)) → MARK(U12(tt, N, XS))
ACTIVE(U11(tt, N, XS)) → U12¹(tt, N, XS)
ACTIVE(U12(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
ACTIVE(U12(tt, N, XS)) → SND(splitAt(N, XS))
ACTIVE(U12(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(U21(tt, X)) → MARK(U22(tt, X))
ACTIVE(U21(tt, X)) → U22¹(tt, X)
ACTIVE(U22(tt, X)) → MARK(X)
ACTIVE(U31(tt, N)) → MARK(U32(tt, N))
ACTIVE(U31(tt, N)) → U32¹(tt, N)
ACTIVE(U32(tt, N)) → MARK(N)
ACTIVE(U41(tt, N, XS)) → MARK(U42(tt, N, XS))
ACTIVE(U41(tt, N, XS)) → U42¹(tt, N, XS)
ACTIVE(U42(tt, N, XS)) → MARK(head(afterNth(N, XS)))
ACTIVE(U42(tt, N, XS)) → HEAD(afterNth(N, XS))
ACTIVE(U42(tt, N, XS)) → AFTERNTH(N, XS)
ACTIVE(U51(tt, Y)) → MARK(U52(tt, Y))
ACTIVE(U51(tt, Y)) → U52¹(tt, Y)
ACTIVE(U52(tt, Y)) → MARK(Y)
ACTIVE(U61(tt, N, X, XS)) → MARK(U62(tt, N, X, XS))
ACTIVE(U61(tt, N, X, XS)) → U62¹(tt, N, X, XS)
ACTIVE(U62(tt, N, X, XS)) → MARK(U63(tt, N, X, XS))
ACTIVE(U62(tt, N, X, XS)) → U63¹(tt, N, X, XS)
ACTIVE(U63(tt, N, X, XS)) → MARK(U64(splitAt(N, XS), X))
ACTIVE(U63(tt, N, X, XS)) → U64¹(splitAt(N, XS), X)
ACTIVE(U63(tt, N, X, XS)) → SPLITAT(N, XS)
ACTIVE(U64(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
ACTIVE(U64(pair(YS, ZS), X)) → PAIR(cons(X, YS), ZS)
ACTIVE(U64(pair(YS, ZS), X)) → CONS(X, YS)
ACTIVE(U71(tt, XS)) → MARK(U72(tt, XS))
ACTIVE(U71(tt, XS)) → U72¹(tt, XS)
ACTIVE(U72(tt, XS)) → MARK(XS)
ACTIVE(U81(tt, N, XS)) → MARK(U82(tt, N, XS))
ACTIVE(U81(tt, N, XS)) → U82¹(tt, N, XS)
ACTIVE(U82(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(U82(tt, N, XS)) → FST(splitAt(N, XS))
ACTIVE(U82(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(afterNth(N, XS)) → MARK(U11(tt, N, XS))
ACTIVE(afterNth(N, XS)) → U11¹(tt, N, XS)
ACTIVE(fst(pair(X, Y))) → MARK(U21(tt, X))
ACTIVE(fst(pair(X, Y))) → U21¹(tt, X)
ACTIVE(head(cons(N, XS))) → MARK(U31(tt, N))
ACTIVE(head(cons(N, XS))) → U31¹(tt, N)
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
ACTIVE(natsFrom(N)) → CONS(N, natsFrom(s(N)))
ACTIVE(natsFrom(N)) → NATSFROM(s(N))
ACTIVE(natsFrom(N)) → S(N)
ACTIVE(sel(N, XS)) → MARK(U41(tt, N, XS))
ACTIVE(sel(N, XS)) → U41¹(tt, N, XS)
ACTIVE(snd(pair(X, Y))) → MARK(U51(tt, Y))
ACTIVE(snd(pair(X, Y))) → U51¹(tt, Y)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
ACTIVE(splitAt(0, XS)) → PAIR(nil, XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U61(tt, N, X, XS))
ACTIVE(splitAt(s(N), cons(X, XS))) → U61¹(tt, N, X, XS)
ACTIVE(tail(cons(N, XS))) → MARK(U71(tt, XS))
ACTIVE(tail(cons(N, XS))) → U71¹(tt, XS)
ACTIVE(take(N, XS)) → MARK(U81(tt, N, XS))
ACTIVE(take(N, XS)) → U81¹(tt, N, XS)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → U11¹(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U12(X1, X2, X3)) → U12¹(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(snd(X)) → SND(mark(X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
MARK(splitAt(X1, X2)) → SPLITAT(mark(X1), mark(X2))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → U21¹(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → U22¹(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → U31¹(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
MARK(U32(X1, X2)) → U32¹(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → U41¹(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
MARK(U42(X1, X2, X3)) → U42¹(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(head(X)) → HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(afterNth(X1, X2)) → AFTERNTH(mark(X1), mark(X2))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U51(X1, X2)) → U51¹(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U52(X1, X2)) → U52¹(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3, X4)) → ACTIVE(U61(mark(X1), X2, X3, X4))
MARK(U61(X1, X2, X3, X4)) → U61¹(mark(X1), X2, X3, X4)
MARK(U61(X1, X2, X3, X4)) → MARK(X1)
MARK(U62(X1, X2, X3, X4)) → ACTIVE(U62(mark(X1), X2, X3, X4))
MARK(U62(X1, X2, X3, X4)) → U62¹(mark(X1), X2, X3, X4)
MARK(U62(X1, X2, X3, X4)) → MARK(X1)
MARK(U63(X1, X2, X3, X4)) → ACTIVE(U63(mark(X1), X2, X3, X4))
MARK(U63(X1, X2, X3, X4)) → U63¹(mark(X1), X2, X3, X4)
MARK(U63(X1, X2, X3, X4)) → MARK(X1)
MARK(U64(X1, X2)) → ACTIVE(U64(mark(X1), X2))
MARK(U64(X1, X2)) → U64¹(mark(X1), X2)
MARK(U64(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
MARK(pair(X1, X2)) → PAIR(mark(X1), mark(X2))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U71(X1, X2)) → U71¹(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U72(X1, X2)) → U72¹(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
MARK(U81(X1, X2, X3)) → U81¹(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
MARK(U82(X1, X2, X3)) → U82¹(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(fst(X)) → FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(natsFrom(X)) → NATSFROM(mark(X))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(0) → ACTIVE(0)
MARK(nil) → ACTIVE(nil)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)
SND(mark(X)) → SND(X)
SND(active(X)) → SND(X)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)
HEAD(mark(X)) → HEAD(X)
HEAD(active(X)) → HEAD(X)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, mark(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, mark(X4)) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, active(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, mark(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, mark(X4)) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, active(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, mark(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, mark(X4)) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, active(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)
U64¹(mark(X1), X2) → U64¹(X1, X2)
U64¹(X1, mark(X2)) → U64¹(X1, X2)
U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)
FST(mark(X)) → FST(X)
FST(active(X)) → FST(X)
NATSFROM(mark(X)) → NATSFROM(X)
NATSFROM(active(X)) → NATSFROM(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)
TAIL(mark(X)) → TAIL(X)
TAIL(active(X)) → TAIL(X)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(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 31 SCCs with 64 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(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.

TAKE(X1, mark(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > TAKE₁

Status:

TAKE₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(mark(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

TAKE₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > TAKE₂

Status:

TAKE₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(12) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(13) TRUE

(14) Obligation:

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

TAIL(active(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(active(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAIL(x1)  =  TAIL(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > TAIL₁

Status:

TAIL₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(mark(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > TAIL₁

Status:

TAIL₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(20) TRUE

(21) Obligation:

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

SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  SEL(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > SEL₁

Status:

SEL₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

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

SEL(mark(X1), X2) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(mark(X1), X2) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  SEL(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

SEL₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > SEL₂

Status:

SEL₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) TRUE

(30) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > S₁

Status:

S₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > S₁

Status:

S₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(35) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(36) TRUE

(37) Obligation:

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

NATSFROM(active(X)) → NATSFROM(X)
NATSFROM(mark(X)) → NATSFROM(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(active(X)) → NATSFROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
NATSFROM(x1)  =  NATSFROM(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > NATSFROM₁

Status:

NATSFROM₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

NATSFROM(mark(X)) → NATSFROM(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(mark(X)) → NATSFROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > NATSFROM₁

Status:

NATSFROM₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(42) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(43) TRUE

(44) Obligation:

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

FST(active(X)) → FST(X)
FST(mark(X)) → FST(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(active(X)) → FST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FST(x1)  =  FST(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > FST₁

Status:

FST₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

FST(mark(X)) → FST(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(mark(X)) → FST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > FST₁

Status:

FST₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(49) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(50) TRUE

(51) Obligation:

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

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U82^1₂, mark₁]

Status:

U82^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U82^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x3)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U82^1₁, active₁]

Status:

U82^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

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

U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U82^1₁, active₁]

Status:

U82^1₁: [1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U82^1₃, active₁]

Status:

U82^1₃: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(62) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(63) TRUE

(64) Obligation:

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

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U81^1₂, mark₁]

Status:

U81^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U81^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x3)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U81^1₁, active₁]

Status:

U81^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(70) Obligation:

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

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U81^1₁, active₁]

Status:

U81^1₁: [1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(72) Obligation:

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

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U81^1₃, active₁]

Status:

U81^1₃: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(75) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(76) TRUE

(77) Obligation:

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

U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(X1, mark(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U72^1₁

Status:

U72^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(79) Obligation:

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

U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(mark(X1), X2) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U72^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

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

U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U72^1₂

Status:

U72^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(84) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(85) TRUE

(86) Obligation:

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

U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(X1, mark(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U71^1₁

Status:

U71^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(88) Obligation:

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

U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(89) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X1), X2) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U71^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(90) Obligation:

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

U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(91) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U71^1₂

Status:

U71^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(92) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(93) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(94) TRUE

(95) Obligation:

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

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(96) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > CONS₁

Status:

CONS₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(97) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

CONS₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(99) Obligation:

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

CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > CONS₂

Status:

CONS₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(101) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(102) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(103) TRUE

(104) Obligation:

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

PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(105) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, mark(X2)) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2)  =  PAIR(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > PAIR₁

Status:

PAIR₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(106) Obligation:

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

PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(mark(X1), X2) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2)  =  PAIR(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

PAIR₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(108) Obligation:

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

PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(109) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > PAIR₂

Status:

PAIR₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(110) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(111) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(112) TRUE

(113) Obligation:

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

U64¹(X1, mark(X2)) → U64¹(X1, X2)
U64¹(mark(X1), X2) → U64¹(X1, X2)
U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U64¹(X1, mark(X2)) → U64¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U64¹(x1, x2)  =  U64¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U64^1₁

Status:

U64^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(115) Obligation:

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

U64¹(mark(X1), X2) → U64¹(X1, X2)
U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U64¹(mark(X1), X2) → U64¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U64¹(x1, x2)  =  U64¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U64^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(117) Obligation:

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

U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(118) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U64^1₂

Status:

U64^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(119) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(120) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(121) TRUE

(122) Obligation:

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

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, mark(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, mark(X4)) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, active(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, X2, mark(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, mark(X4)) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U63^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(124) Obligation:

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

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, active(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(125) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, X2, active(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U63^1₂, mark₁]

Status:

U63^1₂: multiset
mark₁: multiset
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(126) Obligation:

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

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(127) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x2, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U63^1₂, active₁]

Status:

U63^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(128) Obligation:

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

U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(129) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U63^1₂

Status:

U63^1₂: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(130) Obligation:

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

U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x1, x2, x3)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U63^1₃, mark₁]

Status:

U63^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(132) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(133) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(134) TRUE

(135) Obligation:

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

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, mark(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, mark(X4)) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, active(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, X2, mark(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, mark(X4)) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U62^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(137) Obligation:

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

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, active(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(138) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, X2, active(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U62^1₂, mark₁]

Status:

U62^1₂: multiset
mark₁: multiset
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(139) Obligation:

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

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x2, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U62^1₂, active₁]

Status:

U62^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(141) Obligation:

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

U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(142) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U62^1₂

Status:

U62^1₂: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(143) Obligation:

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

U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(144) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x1, x2, x3)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U62^1₃, mark₁]

Status:

U62^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(145) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(146) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(147) TRUE

(148) Obligation:

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

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, mark(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, mark(X4)) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, active(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, X2, mark(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, mark(X4)) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U61^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(150) Obligation:

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

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, active(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(151) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, X2, active(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U61^1₂, mark₁]

Status:

U61^1₂: multiset
mark₁: multiset
active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(152) Obligation:

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

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(153) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x2, x4)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U61^1₂, active₁]

Status:

U61^1₂: multiset
mark₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(154) Obligation:

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

U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(155) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U61^1₂

Status:

U61^1₂: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(156) Obligation:

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

U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(157) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x1, x2, x3)
mark(x1)  =  mark(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U61^1₃, mark₁]

Status:

U61^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(158) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(159) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(160) TRUE

(161) Obligation:

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

U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(162) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(X1, mark(X2)) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  U52¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U52^1₁

Status:

U52^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(163) Obligation:

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

U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(164) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(mark(X1), X2) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  U52¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U52^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(165) Obligation:

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

U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(166) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U52^1₂

Status:

U52^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(167) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(168) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(169) TRUE

(170) Obligation:

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

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, mark(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U51^1₁

Status:

U51^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(172) Obligation:

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

U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(173) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U51^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(174) Obligation:

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

U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(175) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U51^1₂

Status:

U51^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(176) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(177) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(178) TRUE

(179) Obligation:

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

AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(180) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTERNTH(x1, x2)  =  AFTERNTH(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > AFTERNTH₁

Status:

AFTERNTH₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(181) Obligation:

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

AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(182) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTERNTH(x1, x2)  =  AFTERNTH(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

AFTERNTH₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(183) Obligation:

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

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(184) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > AFTERNTH₂

Status:

AFTERNTH₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(185) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(186) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(187) TRUE

(188) Obligation:

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

HEAD(active(X)) → HEAD(X)
HEAD(mark(X)) → HEAD(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(189) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(active(X)) → HEAD(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
HEAD(x1)  =  HEAD(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > HEAD₁

Status:

HEAD₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(190) Obligation:

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

HEAD(mark(X)) → HEAD(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(191) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(mark(X)) → HEAD(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > HEAD₁

Status:

HEAD₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(192) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(193) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(194) TRUE

(195) Obligation:

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

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(196) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U42^1₂, mark₁]

Status:

U42^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(197) Obligation:

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

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(198) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U42^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(199) Obligation:

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

U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(200) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x3)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U42^1₁, active₁]

Status:

U42^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(201) Obligation:

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

U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(202) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U42^1₁, active₁]

Status:

U42^1₁: [1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(203) Obligation:

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

U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(204) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U42^1₃, active₁]

Status:

U42^1₃: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(205) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(206) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(207) TRUE

(208) Obligation:

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

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(209) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U41^1₂, mark₁]

Status:

U41^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(210) Obligation:

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

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(211) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U41^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(212) Obligation:

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

U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(213) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x3)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U41^1₁, active₁]

Status:

U41^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(214) Obligation:

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

U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(215) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U41^1₁, active₁]

Status:

U41^1₁: [1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(216) Obligation:

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

U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(217) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U41^1₃, active₁]

Status:

U41^1₃: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(218) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(219) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(220) TRUE

(221) Obligation:

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

U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(222) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, mark(X2)) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  U32¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U32^1₁

Status:

U32^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(223) Obligation:

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

U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(224) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(mark(X1), X2) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  U32¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U32^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(225) Obligation:

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

U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(226) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U32^1₂

Status:

U32^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(227) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(228) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(229) TRUE

(230) Obligation:

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

U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(231) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, mark(X2)) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  U31¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U31^1₁

Status:

U31^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(232) Obligation:

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

U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(233) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  U31¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U31^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(234) Obligation:

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

U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(235) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U31^1₂

Status:

U31^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(236) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(237) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(238) TRUE

(239) Obligation:

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

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(240) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, mark(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U22^1₁

Status:

U22^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(241) Obligation:

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

U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(242) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(mark(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U22^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(243) Obligation:

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

U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(244) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U22^1₂

Status:

U22^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(245) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(246) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(247) TRUE

(248) Obligation:

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

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(249) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, mark(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > U21^1₁

Status:

U21^1₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(250) Obligation:

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

U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(251) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U21^1₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(252) Obligation:

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

U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(253) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > U21^1₂

Status:

U21^1₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(254) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(255) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(256) TRUE

(257) Obligation:

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

SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(258) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  SPLITAT(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > SPLITAT₁

Status:

SPLITAT₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(259) Obligation:

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

SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(260) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  SPLITAT(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

SPLITAT₂: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(261) Obligation:

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

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(262) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > SPLITAT₂

Status:

SPLITAT₂: [2,1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(263) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(264) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(265) TRUE

(266) Obligation:

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

SND(active(X)) → SND(X)
SND(mark(X)) → SND(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(267) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(active(X)) → SND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SND(x1)  =  SND(x1)
active(x1)  =  active(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

active₁ > SND₁

Status:

SND₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(268) Obligation:

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

SND(mark(X)) → SND(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(269) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(mark(X)) → SND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > SND₁

Status:

SND₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(270) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(271) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(272) TRUE

(273) Obligation:

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

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(274) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U12^1₂, mark₁]

Status:

U12^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(275) Obligation:

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

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(276) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U12^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(277) Obligation:

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

U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(278) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x3)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U12^1₁, active₁]

Status:

U12^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(279) Obligation:

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

U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(280) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U12^1₁, active₁]

Status:

U12^1₁: [1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(281) Obligation:

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

U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(282) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U12^1₃, active₁]

Status:

U12^1₃: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(283) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(284) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(285) TRUE

(286) Obligation:

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

U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(287) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

[U11^1₂, mark₁]

Status:

U11^1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(288) Obligation:

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

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(289) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

U11^1₃: [3,2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(290) Obligation:

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

U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(291) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x3)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U11^1₁, active₁]

Status:

U11^1₁: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(292) Obligation:

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

U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(293) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x2)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[U11^1₁, active₁]

Status:

U11^1₁: [1]
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(294) Obligation:

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

U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(295) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:

[U11^1₃, active₁]

Status:

U11^1₃: multiset
active₁: multiset

The following usable rules [FROCOS05] were oriented: none

(296) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(297) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(298) TRUE

(299) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, N, XS)) → MARK(U12(tt, N, XS))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(U21(tt, X)) → MARK(U22(tt, X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(U22(tt, X)) → MARK(X)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U31(tt, N)) → MARK(U32(tt, N))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U32(tt, N)) → MARK(N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U41(tt, N, XS)) → MARK(U42(tt, N, XS))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(U42(tt, N, XS)) → MARK(head(afterNth(N, XS)))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
ACTIVE(U51(tt, Y)) → MARK(U52(tt, Y))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
ACTIVE(U52(tt, Y)) → MARK(Y)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(U61(tt, N, X, XS)) → MARK(U62(tt, N, X, XS))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(U62(tt, N, X, XS)) → MARK(U63(tt, N, X, XS))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U63(tt, N, X, XS)) → MARK(U64(splitAt(N, XS), X))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U64(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3, X4)) → ACTIVE(U61(mark(X1), X2, X3, X4))
ACTIVE(U71(tt, XS)) → MARK(U72(tt, XS))
MARK(U61(X1, X2, X3, X4)) → MARK(X1)
MARK(U62(X1, X2, X3, X4)) → ACTIVE(U62(mark(X1), X2, X3, X4))
ACTIVE(U72(tt, XS)) → MARK(XS)
MARK(U62(X1, X2, X3, X4)) → MARK(X1)
MARK(U63(X1, X2, X3, X4)) → ACTIVE(U63(mark(X1), X2, X3, X4))
ACTIVE(U81(tt, N, XS)) → MARK(U82(tt, N, XS))
MARK(U63(X1, X2, X3, X4)) → MARK(X1)
MARK(U64(X1, X2)) → ACTIVE(U64(mark(X1), X2))
ACTIVE(U82(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(U64(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
ACTIVE(afterNth(N, XS)) → MARK(U11(tt, N, XS))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(fst(pair(X, Y))) → MARK(U21(tt, X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(head(cons(N, XS))) → MARK(U31(tt, N))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
ACTIVE(sel(N, XS)) → MARK(U41(tt, N, XS))
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
ACTIVE(snd(pair(X, Y))) → MARK(U51(tt, Y))
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(fst(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U61(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(tail(cons(N, XS))) → MARK(U71(tt, XS))
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(U81(tt, N, XS))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(300) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  U12
snd(x1)  =  snd
splitAt(x1, x2)  =  splitAt
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
head(x1)  =  head
afterNth(x1, x2)  =  afterNth
U51(x1, x2)  =  U51
U52(x1, x2)  =  U52
U61(x1, x2, x3, x4)  =  U61
U62(x1, x2, x3, x4)  =  U62
U63(x1, x2, x3, x4)  =  U63
U64(x1, x2)  =  U64
pair(x1, x2)  =  pair
cons(x1, x2)  =  cons
U71(x1, x2)  =  U71
U72(x1, x2)  =  U72
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
fst(x1)  =  fst
natsFrom(x1)  =  natsFrom
s(x1)  =  s
sel(x1, x2)  =  sel
0  =  0
nil  =  nil
tail(x1)  =  tail
take(x1, x2)  =  take
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[MARK, U11, tt, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, cons, U71, U72, U81, U82, fst, natsFrom, s, sel, 0, tail, take] > active₁ > pair
[MARK, U11, tt, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, cons, U71, U72, U81, U82, fst, natsFrom, s, sel, 0, tail, take] > active₁ > nil

Status:

MARK: multiset
U11: multiset
tt: multiset
U12: multiset
snd: multiset
splitAt: multiset
U21: multiset
U22: multiset
U31: multiset
U32: multiset
U41: multiset
U42: multiset
head: multiset
afterNth: multiset
U51: multiset
U52: multiset
U61: multiset
U62: multiset
U63: multiset
U64: multiset
pair: []
cons: multiset
U71: multiset
U72: multiset
U81: multiset
U82: multiset
fst: multiset
natsFrom: multiset
s: multiset
sel: multiset
0: multiset
nil: multiset
tail: multiset
take: multiset
active₁: [1]

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
snd(active(X)) → snd(X)
snd(mark(X)) → snd(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
head(active(X)) → head(X)
head(mark(X)) → head(X)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U52(X1, mark(X2)) → U52(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U51(X1, mark(X2)) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(mark(X1), X2) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(active(X)) → fst(X)
fst(mark(X)) → fst(X)
U71(X1, mark(X2)) → U71(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
natsFrom(active(X)) → natsFrom(X)
natsFrom(mark(X)) → natsFrom(X)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(active(X)) → tail(X)
tail(mark(X)) → tail(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(301) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, N, XS)) → MARK(U12(tt, N, XS))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(U21(tt, X)) → MARK(U22(tt, X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(U22(tt, X)) → MARK(X)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U31(tt, N)) → MARK(U32(tt, N))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U32(tt, N)) → MARK(N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U41(tt, N, XS)) → MARK(U42(tt, N, XS))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(U42(tt, N, XS)) → MARK(head(afterNth(N, XS)))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
ACTIVE(U51(tt, Y)) → MARK(U52(tt, Y))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
ACTIVE(U52(tt, Y)) → MARK(Y)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(U61(tt, N, X, XS)) → MARK(U62(tt, N, X, XS))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(U62(tt, N, X, XS)) → MARK(U63(tt, N, X, XS))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U63(tt, N, X, XS)) → MARK(U64(splitAt(N, XS), X))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U64(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3, X4)) → ACTIVE(U61(mark(X1), X2, X3, X4))
ACTIVE(U71(tt, XS)) → MARK(U72(tt, XS))
MARK(U61(X1, X2, X3, X4)) → MARK(X1)
MARK(U62(X1, X2, X3, X4)) → ACTIVE(U62(mark(X1), X2, X3, X4))
ACTIVE(U72(tt, XS)) → MARK(XS)
MARK(U62(X1, X2, X3, X4)) → MARK(X1)
MARK(U63(X1, X2, X3, X4)) → ACTIVE(U63(mark(X1), X2, X3, X4))
ACTIVE(U81(tt, N, XS)) → MARK(U82(tt, N, XS))
MARK(U63(X1, X2, X3, X4)) → MARK(X1)
MARK(U64(X1, X2)) → ACTIVE(U64(mark(X1), X2))
ACTIVE(U82(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(U64(X1, X2)) → MARK(X1)
ACTIVE(afterNth(N, XS)) → MARK(U11(tt, N, XS))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(fst(pair(X, Y))) → MARK(U21(tt, X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(head(cons(N, XS))) → MARK(U31(tt, N))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
ACTIVE(sel(N, XS)) → MARK(U41(tt, N, XS))
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
ACTIVE(snd(pair(X, Y))) → MARK(U51(tt, Y))
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(fst(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U61(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(tail(cons(N, XS))) → MARK(U71(tt, XS))
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(U81(tt, N, XS))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(302) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  x1
tt  =  tt
U12(x1, x2, x3)  =  U12
snd(x1)  =  snd
splitAt(x1, x2)  =  splitAt
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
head(x1)  =  head
afterNth(x1, x2)  =  afterNth
U51(x1, x2)  =  U51
U52(x1, x2)  =  U52
U61(x1, x2, x3, x4)  =  U61
U62(x1, x2, x3, x4)  =  U62
U63(x1, x2, x3, x4)  =  U63
U64(x1, x2)  =  U64
pair(x1, x2)  =  pair(x1, x2)
cons(x1, x2)  =  cons
U71(x1, x2)  =  U71
U72(x1, x2)  =  U72
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
fst(x1)  =  fst
natsFrom(x1)  =  natsFrom
s(x1)  =  s
sel(x1, x2)  =  sel
0  =  0
nil  =  nil
tail(x1)  =  tail
take(x1, x2)  =  take
active(x1)  =  active

Recursive path order with status [RPO].
Quasi-Precedence:

[MARK, U11, tt, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, U71, U72, U81, U82, fst, natsFrom, s, sel, 0, tail, take] > [cons, active] > pair₂
nil > [cons, active] > pair₂

Status:

MARK: []
U11: []
tt: multiset
U12: []
snd: []
splitAt: []
U21: []
U22: []
U31: []
U32: []
U41: []
U42: []
head: []
afterNth: []
U51: []
U52: []
U61: []
U62: []
U63: []
U64: []
pair₂: multiset
cons: multiset
U71: []
U72: []
U81: []
U82: []
fst: []
natsFrom: []
s: []
sel: []
0: multiset
nil: multiset
tail: []
take: []
active: multiset

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
snd(active(X)) → snd(X)
snd(mark(X)) → snd(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
head(active(X)) → head(X)
head(mark(X)) → head(X)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U52(X1, mark(X2)) → U52(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U51(X1, mark(X2)) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(mark(X1), X2) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(active(X)) → fst(X)
fst(mark(X)) → fst(X)
U71(X1, mark(X2)) → U71(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
natsFrom(active(X)) → natsFrom(X)
natsFrom(mark(X)) → natsFrom(X)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(active(X)) → tail(X)
tail(mark(X)) → tail(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(303) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, N, XS)) → MARK(U12(tt, N, XS))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(U21(tt, X)) → MARK(U22(tt, X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(U22(tt, X)) → MARK(X)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U31(tt, N)) → MARK(U32(tt, N))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U32(tt, N)) → MARK(N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U41(tt, N, XS)) → MARK(U42(tt, N, XS))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(U42(tt, N, XS)) → MARK(head(afterNth(N, XS)))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
ACTIVE(U51(tt, Y)) → MARK(U52(tt, Y))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
ACTIVE(U52(tt, Y)) → MARK(Y)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(U61(tt, N, X, XS)) → MARK(U62(tt, N, X, XS))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(U62(tt, N, X, XS)) → MARK(U63(tt, N, X, XS))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U63(tt, N, X, XS)) → MARK(U64(splitAt(N, XS), X))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U64(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3, X4)) → ACTIVE(U61(mark(X1), X2, X3, X4))
ACTIVE(U71(tt, XS)) → MARK(U72(tt, XS))
MARK(U61(X1, X2, X3, X4)) → MARK(X1)
MARK(U62(X1, X2, X3, X4)) → ACTIVE(U62(mark(X1), X2, X3, X4))
ACTIVE(U72(tt, XS)) → MARK(XS)
MARK(U62(X1, X2, X3, X4)) → MARK(X1)
MARK(U63(X1, X2, X3, X4)) → ACTIVE(U63(mark(X1), X2, X3, X4))
ACTIVE(U81(tt, N, XS)) → MARK(U82(tt, N, XS))
MARK(U63(X1, X2, X3, X4)) → MARK(X1)
MARK(U64(X1, X2)) → ACTIVE(U64(mark(X1), X2))
ACTIVE(U82(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(U64(X1, X2)) → MARK(X1)
ACTIVE(afterNth(N, XS)) → MARK(U11(tt, N, XS))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
ACTIVE(fst(pair(X, Y))) → MARK(U21(tt, X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(head(cons(N, XS))) → MARK(U31(tt, N))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
ACTIVE(sel(N, XS)) → MARK(U41(tt, N, XS))
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
ACTIVE(snd(pair(X, Y))) → MARK(U51(tt, Y))
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(fst(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U61(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(tail(cons(N, XS))) → MARK(U71(tt, XS))
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(U81(tt, N, XS))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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

(304) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK
U11(x1, x2, x3)  =  U11
ACTIVE(x1)  =  x1
mark(x1)  =  mark(x1)
tt  =  tt
U12(x1, x2, x3)  =  U12
snd(x1)  =  snd
splitAt(x1, x2)  =  splitAt
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
head(x1)  =  head
afterNth(x1, x2)  =  afterNth
U51(x1, x2)  =  U51
U52(x1, x2)  =  U52
U61(x1, x2, x3, x4)  =  U61
U62(x1, x2, x3, x4)  =  U62
U63(x1, x2, x3, x4)  =  U63
U64(x1, x2)  =  U64
pair(x1, x2)  =  pair
cons(x1, x2)  =  cons(x2)
U71(x1, x2)  =  U71
U72(x1, x2)  =  U72
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
fst(x1)  =  fst
natsFrom(x1)  =  natsFrom
s(x1)  =  s
sel(x1, x2)  =  sel
0  =  0
nil  =  nil
tail(x1)  =  tail
take(x1, x2)  =  take
active(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

mark₁ > [MARK, U11, tt, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, U71, U72, U81, U82, fst, natsFrom, sel, 0, nil, tail, take] > pair
mark₁ > [MARK, U11, tt, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, U71, U72, U81, U82, fst, natsFrom, sel, 0, nil, tail, take] > cons₁
mark₁ > [MARK, U11, tt, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, U71, U72, U81, U82, fst, natsFrom, sel, 0, nil, tail, take] > s

Status:

MARK: []
U11: []
mark₁: multiset
tt: multiset
U12: []
snd: []
splitAt: []
U21: []
U22: []
U31: []
U32: []
U41: []
U42: []
head: []
afterNth: []
U51: []
U52: []
U61: []
U62: []
U63: []
U64: []
pair: multiset
cons₁: [1]
U71: []
U72: []
U81: []
U82: []
fst: []
natsFrom: []
s: []
sel: []
0: multiset
nil: multiset
tail: []
take: []

The following usable rules [FROCOS05] were oriented:

U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
snd(active(X)) → snd(X)
snd(mark(X)) → snd(X)
U22(X1, mark(X2)) → U22(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
head(active(X)) → head(X)
head(mark(X)) → head(X)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U52(X1, mark(X2)) → U52(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U51(X1, mark(X2)) → U51(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(mark(X1), X2) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U72(X1, mark(X2)) → U72(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(active(X)) → fst(X)
fst(mark(X)) → fst(X)
U71(X1, mark(X2)) → U71(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
natsFrom(active(X)) → natsFrom(X)
natsFrom(mark(X)) → natsFrom(X)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(active(X)) → tail(X)
tail(mark(X)) → tail(X)
take(X1, mark(X2)) → take(X1, X2)
take(mark(X1), X2) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

(305) Obligation:

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

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, N, XS)) → MARK(U12(tt, N, XS))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(U21(tt, X)) → MARK(U22(tt, X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(U22(tt, X)) → MARK(X)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U31(tt, N)) → MARK(U32(tt, N))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U32(tt, N)) → MARK(N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U41(tt, N, XS)) → MARK(U42(tt, N, XS))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(U42(tt, N, XS)) → MARK(head(afterNth(N, XS)))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
ACTIVE(U51(tt, Y)) → MARK(U52(tt, Y))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
ACTIVE(U52(tt, Y)) → MARK(Y)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(U61(tt, N, X, XS)) → MARK(U62(tt, N, X, XS))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(U62(tt, N, X, XS)) → MARK(U63(tt, N, X, XS))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U63(tt, N, X, XS)) → MARK(U64(splitAt(N, XS), X))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U64(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3, X4)) → ACTIVE(U61(mark(X1), X2, X3, X4))
ACTIVE(U71(tt, XS)) → MARK(U72(tt, XS))
MARK(U61(X1, X2, X3, X4)) → MARK(X1)
MARK(U62(X1, X2, X3, X4)) → ACTIVE(U62(mark(X1), X2, X3, X4))
ACTIVE(U72(tt, XS)) → MARK(XS)
MARK(U62(X1, X2, X3, X4)) → MARK(X1)
MARK(U63(X1, X2, X3, X4)) → ACTIVE(U63(mark(X1), X2, X3, X4))
ACTIVE(U81(tt, N, XS)) → MARK(U82(tt, N, XS))
MARK(U63(X1, X2, X3, X4)) → MARK(X1)
MARK(U64(X1, X2)) → ACTIVE(U64(mark(X1), X2))
ACTIVE(U82(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(U64(X1, X2)) → MARK(X1)
ACTIVE(afterNth(N, XS)) → MARK(U11(tt, N, XS))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
ACTIVE(fst(pair(X, Y))) → MARK(U21(tt, X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(head(cons(N, XS))) → MARK(U31(tt, N))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
ACTIVE(sel(N, XS)) → MARK(U41(tt, N, XS))
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
ACTIVE(snd(pair(X, Y))) → MARK(U51(tt, Y))
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(fst(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U61(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
ACTIVE(tail(cons(N, XS))) → MARK(U71(tt, XS))
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(U81(tt, N, XS))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

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