(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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > TAKE2

Status:
TAKE2: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(7) 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.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


TAKE(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > TAKE1

Status:
TAKE1: [1]
active1: [1]


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)

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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[TAKE2, active1]

Status:
TAKE2: [2,1]
active1: [1]


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[TAIL1, active1]

Status:
TAIL1: [1]
active1: [1]


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: Combined order from the following AFS and order.
TAIL(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > SEL2

Status:
SEL2: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(23) 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.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


SEL(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > SEL1

Status:
SEL1: [1]
active1: [1]


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)

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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[SEL2, active1]

Status:
SEL2: [2,1]
active1: [1]


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[S1, active1]

Status:
S1: [1]
active1: [1]


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: Combined order from the following AFS and order.
S(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[NATSFROM1, active1]

Status:
NATSFROM1: [1]
active1: [1]


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: Combined order from the following AFS and order.
NATSFROM(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[FST1, active1]

Status:
FST1: [1]
active1: [1]


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: Combined order from the following AFS and order.
FST(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


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, 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(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U82^11: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U82^13

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)  =  x3
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


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, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U82^12, active1]

Status:
U82^12: [2,1]
active1: [1]


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: Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U82^13

Status:
U82^13: [3,2,1]
active1: [1]


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, 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(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U81^11: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U81^13

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)  =  x3
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


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, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U81^12, active1]

Status:
U81^12: [2,1]
active1: [1]


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: Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U81^13

Status:
U81^13: [3,2,1]
active1: [1]


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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U72^12

Status:
U72^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(79) 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.

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U721(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U72^11

Status:
U72^11: [1]
active1: [1]


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)

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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U72^12, active1]

Status:
U72^12: [2,1]
active1: [1]


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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U71^12

Status:
U71^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(88) 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.

(89) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U711(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U71^11

Status:
U71^11: [1]
active1: [1]


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)

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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U71^12, active1]

Status:
U71^12: [2,1]
active1: [1]


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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > CONS2

Status:
CONS2: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(97) 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.

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > CONS1

Status:
CONS1: [1]
active1: [1]


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)

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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[CONS2, active1]

Status:
CONS2: [2,1]
active1: [1]


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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > PAIR2

Status:
PAIR2: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(106) 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.

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PAIR(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > PAIR1

Status:
PAIR1: [1]
active1: [1]


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)

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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[PAIR2, active1]

Status:
PAIR2: [2,1]
active1: [1]


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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U64^12

Status:
U64^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(115) 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.

(116) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U641(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U64^11

Status:
U64^11: [1]
active1: [1]


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)

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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U64^12, active1]

Status:
U64^12: [2,1]
active1: [1]


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, 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(x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U63^11: [1]
mark1: [1]


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(X1, X2, mark(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, mark(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(x3, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U63^12, mark1]

Status:
U63^12: [2,1]
mark1: [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)
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.

(127) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  x4
mark(x1)  =  mark
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark: []
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(128) 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)

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(X1, X2, active(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(x3, x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U63^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(130) 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.

(131) 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)

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > [U63^12, active1]

Status:
U63^12: [1,2]
mark1: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(132) 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.

(133) 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)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(134) Obligation:

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

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.

(135) 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)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(136) 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.

(137) PisEmptyProof (EQUIVALENT transformation)

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

(138) TRUE

(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(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.

(140) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U62^11: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(141) 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(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.

(142) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U62^12, mark1]

Status:
U62^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(143) 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.

(144) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  x4
mark(x1)  =  mark
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark: []
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(145) 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)

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) 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)
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)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U62^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(147) 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.

(148) 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)

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > [U62^12, active1]

Status:
U62^12: [1,2]
mark1: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(149) 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.

(150) 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)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(151) Obligation:

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

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.

(152) 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)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(153) 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.

(154) PisEmptyProof (EQUIVALENT transformation)

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

(155) TRUE

(156) 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.

(157) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U61^11: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(158) 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(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.

(159) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U61^12, mark1]

Status:
U61^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(160) 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.

(161) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)  =  x4
mark(x1)  =  mark
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark: []
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(162) 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)

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.

(163) 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)
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)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U61^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(164) 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.

(165) 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)

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > [U61^12, active1]

Status:
U61^12: [1,2]
mark1: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(166) 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.

(167) 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)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(168) Obligation:

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

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.

(169) 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)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(170) 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.

(171) PisEmptyProof (EQUIVALENT transformation)

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

(172) TRUE

(173) 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.

(174) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U52^12

Status:
U52^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(175) 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.

(176) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U521(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U52^11

Status:
U52^11: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(177) Obligation:

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

U521(active(X1), 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.

(178) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U52^12, active1]

Status:
U52^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(179) 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.

(180) PisEmptyProof (EQUIVALENT transformation)

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

(181) TRUE

(182) 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.

(183) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U51^12

Status:
U51^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(184) 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.

(185) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U51^11

Status:
U51^11: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(186) Obligation:

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

U511(active(X1), 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.

(187) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U51^12, active1]

Status:
U51^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(188) 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.

(189) PisEmptyProof (EQUIVALENT transformation)

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

(190) TRUE

(191) 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.

(192) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > AFTERNTH2

Status:
AFTERNTH2: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(193) 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.

(194) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


AFTERNTH(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > AFTERNTH1

Status:
AFTERNTH1: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(195) Obligation:

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

AFTERNTH(active(X1), 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.

(196) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[AFTERNTH2, active1]

Status:
AFTERNTH2: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(197) 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.

(198) PisEmptyProof (EQUIVALENT transformation)

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

(199) TRUE

(200) 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.

(201) 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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[HEAD1, active1]

Status:
HEAD1: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(202) 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.

(203) 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: Combined order from the following AFS and order.
HEAD(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(204) 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.

(205) PisEmptyProof (EQUIVALENT transformation)

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

(206) TRUE

(207) 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.

(208) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U42^11: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(209) 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(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.

(210) 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(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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U42^13

Status:
U42^13: [3,2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(211) 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.

(212) 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)  =  x3
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(213) 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.

(214) 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, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U42^12, active1]

Status:
U42^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(215) 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.

(216) 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: Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U42^13

Status:
U42^13: [3,2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(217) 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.

(218) PisEmptyProof (EQUIVALENT transformation)

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

(219) TRUE

(220) 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.

(221) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U41^11: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(222) 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(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.

(223) 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(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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U41^13

Status:
U41^13: [3,2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(224) 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.

(225) 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)  =  x3
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(226) 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.

(227) 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, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U41^12, active1]

Status:
U41^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(228) 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.

(229) 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: Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U41^13

Status:
U41^13: [3,2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(230) 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.

(231) PisEmptyProof (EQUIVALENT transformation)

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

(232) TRUE

(233) 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.

(234) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U32^12

Status:
U32^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(235) 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.

(236) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U321(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U32^11

Status:
U32^11: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(237) Obligation:

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

U321(active(X1), 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.

(238) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U32^12, active1]

Status:
U32^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(239) 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.

(240) PisEmptyProof (EQUIVALENT transformation)

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

(241) TRUE

(242) 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.

(243) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U31^12

Status:
U31^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(244) 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.

(245) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U31^11

Status:
U31^11: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(246) Obligation:

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

U311(active(X1), 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.

(247) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U31^12, active1]

Status:
U31^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(248) 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.

(249) PisEmptyProof (EQUIVALENT transformation)

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

(250) TRUE

(251) 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.

(252) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U22^12

Status:
U22^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(253) 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.

(254) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U221(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U22^11

Status:
U22^11: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(255) Obligation:

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

U221(active(X1), 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.

(256) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U22^12, active1]

Status:
U22^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(257) 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.

(258) PisEmptyProof (EQUIVALENT transformation)

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

(259) TRUE

(260) 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.

(261) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U21^12

Status:
U21^12: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(262) 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.

(263) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U21^11

Status:
U21^11: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(264) Obligation:

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

U211(active(X1), 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.

(265) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[U21^12, active1]

Status:
U21^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(266) 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.

(267) PisEmptyProof (EQUIVALENT transformation)

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

(268) TRUE

(269) 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.

(270) 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)
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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > SPLITAT2

Status:
SPLITAT2: [2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(271) 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.

(272) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


SPLITAT(X1, active(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)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > SPLITAT1

Status:
SPLITAT1: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(273) Obligation:

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

SPLITAT(active(X1), 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.

(274) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
[SPLITAT2, active1]

Status:
SPLITAT2: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(275) 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.

(276) PisEmptyProof (EQUIVALENT transformation)

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

(277) TRUE

(278) 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.

(279) 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

Lexicographic path order with status [LPO].
Quasi-Precedence:
[SND1, active1]

Status:
SND1: [1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(280) 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.

(281) 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: Combined order from the following AFS and order.
SND(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(282) 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.

(283) PisEmptyProof (EQUIVALENT transformation)

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

(284) TRUE

(285) 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.

(286) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U12^11: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(287) 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(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.

(288) 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(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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U12^13

Status:
U12^13: [3,2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(289) 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.

(290) 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)  =  x3
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(291) 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.

(292) 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, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U12^12, active1]

Status:
U12^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(293) 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.

(294) 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: Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U12^13

Status:
U12^13: [3,2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(295) 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.

(296) PisEmptyProof (EQUIVALENT transformation)

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

(297) TRUE

(298) 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.

(299) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
U11^11: [1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(300) 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(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.

(301) 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(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

Lexicographic path order with status [LPO].
Quasi-Precedence:
mark1 > U11^13

Status:
U11^13: [3,2,1]
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(302) 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.

(303) 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)  =  x3
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial

Status:
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(304) 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.

(305) 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, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:
[U11^12, active1]

Status:
U11^12: [2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(306) 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.

(307) 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: Lexicographic path order with status [LPO].
Quasi-Precedence:
active1 > U11^13

Status:
U11^13: [3,2,1]
active1: [1]


The following usable rules [FROCOS05] were oriented: none

(308) 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.

(309) PisEmptyProof (EQUIVALENT transformation)

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

(310) TRUE

(311) 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.

(312) 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
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)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[MARK, U11, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, pair, U71, U72, U81, U82, fst, natsFrom, s, sel, tail, take] > tt > cons
[MARK, U11, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, pair, U71, U72, U81, U82, fst, natsFrom, s, sel, tail, take] > nil > cons
0 > nil > cons

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

(313) 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)
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.

(314) 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)  =  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(x1, 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

Lexicographic path order with status [LPO].
Quasi-Precedence:
tt > [MARK, U11, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, pair, cons2, U71, U72, U81, U82, fst, natsFrom, sel, tail, take] > s
0 > [MARK, U11, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, pair, cons2, U71, U72, U81, U82, fst, natsFrom, sel, tail, take] > s
0 > nil

Status:
MARK: []
U11: []
tt: []
U12: []
snd: []
splitAt: []
U21: []
U22: []
U31: []
U32: []
U41: []
U42: []
head: []
afterNth: []
U51: []
U52: []
U61: []
U62: []
U63: []
U64: []
pair: []
cons2: [2,1]
U71: []
U72: []
U81: []
U82: []
fst: []
natsFrom: []
s: []
sel: []
0: []
nil: []
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)
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)

(315) 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)
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.

(316) 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(x1, 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(x1)
sel(x1, x2)  =  sel
0  =  0
nil  =  nil
tail(x1)  =  tail
take(x1, x2)  =  take
active(x1)  =  active

Lexicographic path order with status [LPO].
Quasi-Precedence:
active > [0, nil] > [MARK, U11, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, cons2, U71, U72, U81, U82, fst, natsFrom, sel, tail, take] > tt > s1
active > [0, nil] > [MARK, U11, U12, snd, splitAt, U21, U22, U31, U32, U41, U42, head, afterNth, U51, U52, U61, U62, U63, U64, cons2, U71, U72, U81, U82, fst, natsFrom, sel, tail, take] > pair > s1

Status:
MARK: []
U11: []
tt: []
U12: []
snd: []
splitAt: []
U21: []
U22: []
U31: []
U32: []
U41: []
U42: []
head: []
afterNth: []
U51: []
U52: []
U61: []
U62: []
U63: []
U64: []
pair: []
cons2: [1,2]
U71: []
U72: []
U81: []
U82: []
fst: []
natsFrom: []
s1: [1]
sel: []
0: []
nil: []
tail: []
take: []
active: []


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)
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)
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)

(317) 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.