(0) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

ACTIVE(U101(tt, V2)) → MARK(U102(isLNat(V2)))
ACTIVE(U101(tt, V2)) → U1021(isLNat(V2))
ACTIVE(U101(tt, V2)) → ISLNAT(V2)
ACTIVE(U102(tt)) → MARK(tt)
ACTIVE(U11(tt, N, XS)) → MARK(U12(isLNat(XS), N, XS))
ACTIVE(U11(tt, N, XS)) → U121(isLNat(XS), N, XS)
ACTIVE(U11(tt, N, XS)) → ISLNAT(XS)
ACTIVE(U111(tt)) → MARK(tt)
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(U121(tt)) → MARK(tt)
ACTIVE(U131(tt, V2)) → MARK(U132(isLNat(V2)))
ACTIVE(U131(tt, V2)) → U1321(isLNat(V2))
ACTIVE(U131(tt, V2)) → ISLNAT(V2)
ACTIVE(U132(tt)) → MARK(tt)
ACTIVE(U141(tt, V2)) → MARK(U142(isLNat(V2)))
ACTIVE(U141(tt, V2)) → U1421(isLNat(V2))
ACTIVE(U141(tt, V2)) → ISLNAT(V2)
ACTIVE(U142(tt)) → MARK(tt)
ACTIVE(U151(tt, V2)) → MARK(U152(isLNat(V2)))
ACTIVE(U151(tt, V2)) → U1521(isLNat(V2))
ACTIVE(U151(tt, V2)) → ISLNAT(V2)
ACTIVE(U152(tt)) → MARK(tt)
ACTIVE(U161(tt, N)) → MARK(cons(N, natsFrom(s(N))))
ACTIVE(U161(tt, N)) → CONS(N, natsFrom(s(N)))
ACTIVE(U161(tt, N)) → NATSFROM(s(N))
ACTIVE(U161(tt, N)) → S(N)
ACTIVE(U171(tt, N, XS)) → MARK(U172(isLNat(XS), N, XS))
ACTIVE(U171(tt, N, XS)) → U1721(isLNat(XS), N, XS)
ACTIVE(U171(tt, N, XS)) → ISLNAT(XS)
ACTIVE(U172(tt, N, XS)) → MARK(head(afterNth(N, XS)))
ACTIVE(U172(tt, N, XS)) → HEAD(afterNth(N, XS))
ACTIVE(U172(tt, N, XS)) → AFTERNTH(N, XS)
ACTIVE(U181(tt, Y)) → MARK(U182(isLNat(Y), Y))
ACTIVE(U181(tt, Y)) → U1821(isLNat(Y), Y)
ACTIVE(U181(tt, Y)) → ISLNAT(Y)
ACTIVE(U182(tt, Y)) → MARK(Y)
ACTIVE(U191(tt, XS)) → MARK(pair(nil, XS))
ACTIVE(U191(tt, XS)) → PAIR(nil, XS)
ACTIVE(U201(tt, N, X, XS)) → MARK(U202(isNatural(X), N, X, XS))
ACTIVE(U201(tt, N, X, XS)) → U2021(isNatural(X), N, X, XS)
ACTIVE(U201(tt, N, X, XS)) → ISNATURAL(X)
ACTIVE(U202(tt, N, X, XS)) → MARK(U203(isLNat(XS), N, X, XS))
ACTIVE(U202(tt, N, X, XS)) → U2031(isLNat(XS), N, X, XS)
ACTIVE(U202(tt, N, X, XS)) → ISLNAT(XS)
ACTIVE(U203(tt, N, X, XS)) → MARK(U204(splitAt(N, XS), X))
ACTIVE(U203(tt, N, X, XS)) → U2041(splitAt(N, XS), X)
ACTIVE(U203(tt, N, X, XS)) → SPLITAT(N, XS)
ACTIVE(U204(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
ACTIVE(U204(pair(YS, ZS), X)) → PAIR(cons(X, YS), ZS)
ACTIVE(U204(pair(YS, ZS), X)) → CONS(X, YS)
ACTIVE(U21(tt, X, Y)) → MARK(U22(isLNat(Y), X))
ACTIVE(U21(tt, X, Y)) → U221(isLNat(Y), X)
ACTIVE(U21(tt, X, Y)) → ISLNAT(Y)
ACTIVE(U211(tt, XS)) → MARK(U212(isLNat(XS), XS))
ACTIVE(U211(tt, XS)) → U2121(isLNat(XS), XS)
ACTIVE(U211(tt, XS)) → ISLNAT(XS)
ACTIVE(U212(tt, XS)) → MARK(XS)
ACTIVE(U22(tt, X)) → MARK(X)
ACTIVE(U221(tt, N, XS)) → MARK(U222(isLNat(XS), N, XS))
ACTIVE(U221(tt, N, XS)) → U2221(isLNat(XS), N, XS)
ACTIVE(U221(tt, N, XS)) → ISLNAT(XS)
ACTIVE(U222(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(U222(tt, N, XS)) → FST(splitAt(N, XS))
ACTIVE(U222(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(U31(tt, N, XS)) → MARK(U32(isLNat(XS), N))
ACTIVE(U31(tt, N, XS)) → U321(isLNat(XS), N)
ACTIVE(U31(tt, N, XS)) → ISLNAT(XS)
ACTIVE(U32(tt, N)) → MARK(N)
ACTIVE(U41(tt, V2)) → MARK(U42(isLNat(V2)))
ACTIVE(U41(tt, V2)) → U421(isLNat(V2))
ACTIVE(U41(tt, V2)) → ISLNAT(V2)
ACTIVE(U42(tt)) → MARK(tt)
ACTIVE(U51(tt, V2)) → MARK(U52(isLNat(V2)))
ACTIVE(U51(tt, V2)) → U521(isLNat(V2))
ACTIVE(U51(tt, V2)) → ISLNAT(V2)
ACTIVE(U52(tt)) → MARK(tt)
ACTIVE(U61(tt)) → MARK(tt)
ACTIVE(U71(tt)) → MARK(tt)
ACTIVE(U81(tt)) → MARK(tt)
ACTIVE(U91(tt)) → MARK(tt)
ACTIVE(afterNth(N, XS)) → MARK(U11(isNatural(N), N, XS))
ACTIVE(afterNth(N, XS)) → U111(isNatural(N), N, XS)
ACTIVE(afterNth(N, XS)) → ISNATURAL(N)
ACTIVE(fst(pair(X, Y))) → MARK(U21(isLNat(X), X, Y))
ACTIVE(fst(pair(X, Y))) → U211(isLNat(X), X, Y)
ACTIVE(fst(pair(X, Y))) → ISLNAT(X)
ACTIVE(head(cons(N, XS))) → MARK(U31(isNatural(N), N, XS))
ACTIVE(head(cons(N, XS))) → U311(isNatural(N), N, XS)
ACTIVE(head(cons(N, XS))) → ISNATURAL(N)
ACTIVE(isLNat(nil)) → MARK(tt)
ACTIVE(isLNat(afterNth(V1, V2))) → MARK(U41(isNatural(V1), V2))
ACTIVE(isLNat(afterNth(V1, V2))) → U411(isNatural(V1), V2)
ACTIVE(isLNat(afterNth(V1, V2))) → ISNATURAL(V1)
ACTIVE(isLNat(cons(V1, V2))) → MARK(U51(isNatural(V1), V2))
ACTIVE(isLNat(cons(V1, V2))) → U511(isNatural(V1), V2)
ACTIVE(isLNat(cons(V1, V2))) → ISNATURAL(V1)
ACTIVE(isLNat(fst(V1))) → MARK(U61(isPLNat(V1)))
ACTIVE(isLNat(fst(V1))) → U611(isPLNat(V1))
ACTIVE(isLNat(fst(V1))) → ISPLNAT(V1)
ACTIVE(isLNat(natsFrom(V1))) → MARK(U71(isNatural(V1)))
ACTIVE(isLNat(natsFrom(V1))) → U711(isNatural(V1))
ACTIVE(isLNat(natsFrom(V1))) → ISNATURAL(V1)
ACTIVE(isLNat(snd(V1))) → MARK(U81(isPLNat(V1)))
ACTIVE(isLNat(snd(V1))) → U811(isPLNat(V1))
ACTIVE(isLNat(snd(V1))) → ISPLNAT(V1)
ACTIVE(isLNat(tail(V1))) → MARK(U91(isLNat(V1)))
ACTIVE(isLNat(tail(V1))) → U911(isLNat(V1))
ACTIVE(isLNat(tail(V1))) → ISLNAT(V1)
ACTIVE(isLNat(take(V1, V2))) → MARK(U101(isNatural(V1), V2))
ACTIVE(isLNat(take(V1, V2))) → U1011(isNatural(V1), V2)
ACTIVE(isLNat(take(V1, V2))) → ISNATURAL(V1)
ACTIVE(isNatural(0)) → MARK(tt)
ACTIVE(isNatural(head(V1))) → MARK(U111(isLNat(V1)))
ACTIVE(isNatural(head(V1))) → U1111(isLNat(V1))
ACTIVE(isNatural(head(V1))) → ISLNAT(V1)
ACTIVE(isNatural(s(V1))) → MARK(U121(isNatural(V1)))
ACTIVE(isNatural(s(V1))) → U1211(isNatural(V1))
ACTIVE(isNatural(s(V1))) → ISNATURAL(V1)
ACTIVE(isNatural(sel(V1, V2))) → MARK(U131(isNatural(V1), V2))
ACTIVE(isNatural(sel(V1, V2))) → U1311(isNatural(V1), V2)
ACTIVE(isNatural(sel(V1, V2))) → ISNATURAL(V1)
ACTIVE(isPLNat(pair(V1, V2))) → MARK(U141(isLNat(V1), V2))
ACTIVE(isPLNat(pair(V1, V2))) → U1411(isLNat(V1), V2)
ACTIVE(isPLNat(pair(V1, V2))) → ISLNAT(V1)
ACTIVE(isPLNat(splitAt(V1, V2))) → MARK(U151(isNatural(V1), V2))
ACTIVE(isPLNat(splitAt(V1, V2))) → U1511(isNatural(V1), V2)
ACTIVE(isPLNat(splitAt(V1, V2))) → ISNATURAL(V1)
ACTIVE(natsFrom(N)) → MARK(U161(isNatural(N), N))
ACTIVE(natsFrom(N)) → U1611(isNatural(N), N)
ACTIVE(natsFrom(N)) → ISNATURAL(N)
ACTIVE(sel(N, XS)) → MARK(U171(isNatural(N), N, XS))
ACTIVE(sel(N, XS)) → U1711(isNatural(N), N, XS)
ACTIVE(sel(N, XS)) → ISNATURAL(N)
ACTIVE(snd(pair(X, Y))) → MARK(U181(isLNat(X), Y))
ACTIVE(snd(pair(X, Y))) → U1811(isLNat(X), Y)
ACTIVE(snd(pair(X, Y))) → ISLNAT(X)
ACTIVE(splitAt(0, XS)) → MARK(U191(isLNat(XS), XS))
ACTIVE(splitAt(0, XS)) → U1911(isLNat(XS), XS)
ACTIVE(splitAt(0, XS)) → ISLNAT(XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U201(isNatural(N), N, X, XS))
ACTIVE(splitAt(s(N), cons(X, XS))) → U2011(isNatural(N), N, X, XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → ISNATURAL(N)
ACTIVE(tail(cons(N, XS))) → MARK(U211(isNatural(N), XS))
ACTIVE(tail(cons(N, XS))) → U2111(isNatural(N), XS)
ACTIVE(tail(cons(N, XS))) → ISNATURAL(N)
ACTIVE(take(N, XS)) → MARK(U221(isNatural(N), N, XS))
ACTIVE(take(N, XS)) → U2211(isNatural(N), N, XS)
ACTIVE(take(N, XS)) → ISNATURAL(N)
MARK(U101(X1, X2)) → ACTIVE(U101(mark(X1), X2))
MARK(U101(X1, X2)) → U1011(mark(X1), X2)
MARK(U101(X1, X2)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U102(X)) → ACTIVE(U102(mark(X)))
MARK(U102(X)) → U1021(mark(X))
MARK(U102(X)) → MARK(X)
MARK(isLNat(X)) → ACTIVE(isLNat(X))
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(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(U111(X)) → ACTIVE(U111(mark(X)))
MARK(U111(X)) → U1111(mark(X))
MARK(U111(X)) → MARK(X)
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(U121(X)) → ACTIVE(U121(mark(X)))
MARK(U121(X)) → U1211(mark(X))
MARK(U121(X)) → MARK(X)
MARK(U131(X1, X2)) → ACTIVE(U131(mark(X1), X2))
MARK(U131(X1, X2)) → U1311(mark(X1), X2)
MARK(U131(X1, X2)) → MARK(X1)
MARK(U132(X)) → ACTIVE(U132(mark(X)))
MARK(U132(X)) → U1321(mark(X))
MARK(U132(X)) → MARK(X)
MARK(U141(X1, X2)) → ACTIVE(U141(mark(X1), X2))
MARK(U141(X1, X2)) → U1411(mark(X1), X2)
MARK(U141(X1, X2)) → MARK(X1)
MARK(U142(X)) → ACTIVE(U142(mark(X)))
MARK(U142(X)) → U1421(mark(X))
MARK(U142(X)) → MARK(X)
MARK(U151(X1, X2)) → ACTIVE(U151(mark(X1), X2))
MARK(U151(X1, X2)) → U1511(mark(X1), X2)
MARK(U151(X1, X2)) → MARK(X1)
MARK(U152(X)) → ACTIVE(U152(mark(X)))
MARK(U152(X)) → U1521(mark(X))
MARK(U152(X)) → MARK(X)
MARK(U161(X1, X2)) → ACTIVE(U161(mark(X1), X2))
MARK(U161(X1, X2)) → U1611(mark(X1), X2)
MARK(U161(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
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(U171(X1, X2, X3)) → ACTIVE(U171(mark(X1), X2, X3))
MARK(U171(X1, X2, X3)) → U1711(mark(X1), X2, X3)
MARK(U171(X1, X2, X3)) → MARK(X1)
MARK(U172(X1, X2, X3)) → ACTIVE(U172(mark(X1), X2, X3))
MARK(U172(X1, X2, X3)) → U1721(mark(X1), X2, X3)
MARK(U172(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(U181(X1, X2)) → ACTIVE(U181(mark(X1), X2))
MARK(U181(X1, X2)) → U1811(mark(X1), X2)
MARK(U181(X1, X2)) → MARK(X1)
MARK(U182(X1, X2)) → ACTIVE(U182(mark(X1), X2))
MARK(U182(X1, X2)) → U1821(mark(X1), X2)
MARK(U182(X1, X2)) → MARK(X1)
MARK(U191(X1, X2)) → ACTIVE(U191(mark(X1), X2))
MARK(U191(X1, X2)) → U1911(mark(X1), X2)
MARK(U191(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(nil) → ACTIVE(nil)
MARK(U201(X1, X2, X3, X4)) → ACTIVE(U201(mark(X1), X2, X3, X4))
MARK(U201(X1, X2, X3, X4)) → U2011(mark(X1), X2, X3, X4)
MARK(U201(X1, X2, X3, X4)) → MARK(X1)
MARK(U202(X1, X2, X3, X4)) → ACTIVE(U202(mark(X1), X2, X3, X4))
MARK(U202(X1, X2, X3, X4)) → U2021(mark(X1), X2, X3, X4)
MARK(U202(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatural(X)) → ACTIVE(isNatural(X))
MARK(U203(X1, X2, X3, X4)) → ACTIVE(U203(mark(X1), X2, X3, X4))
MARK(U203(X1, X2, X3, X4)) → U2031(mark(X1), X2, X3, X4)
MARK(U203(X1, X2, X3, X4)) → MARK(X1)
MARK(U204(X1, X2)) → ACTIVE(U204(mark(X1), X2))
MARK(U204(X1, X2)) → U2041(mark(X1), X2)
MARK(U204(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
MARK(U21(X1, X2, X3)) → U211(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → 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(U211(X1, X2)) → ACTIVE(U211(mark(X1), X2))
MARK(U211(X1, X2)) → U2111(mark(X1), X2)
MARK(U211(X1, X2)) → MARK(X1)
MARK(U212(X1, X2)) → ACTIVE(U212(mark(X1), X2))
MARK(U212(X1, X2)) → U2121(mark(X1), X2)
MARK(U212(X1, X2)) → MARK(X1)
MARK(U221(X1, X2, X3)) → ACTIVE(U221(mark(X1), X2, X3))
MARK(U221(X1, X2, X3)) → U2211(mark(X1), X2, X3)
MARK(U221(X1, X2, X3)) → MARK(X1)
MARK(U222(X1, X2, X3)) → ACTIVE(U222(mark(X1), X2, X3))
MARK(U222(X1, X2, X3)) → U2221(mark(X1), X2, X3)
MARK(U222(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(fst(X)) → FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
MARK(U31(X1, X2, X3)) → U311(mark(X1), X2, X3)
MARK(U31(X1, X2, X3)) → 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)) → ACTIVE(U41(mark(X1), X2))
MARK(U41(X1, X2)) → U411(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → ACTIVE(U42(mark(X)))
MARK(U42(X)) → U421(mark(X))
MARK(U42(X)) → MARK(X)
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(X)) → ACTIVE(U52(mark(X)))
MARK(U52(X)) → U521(mark(X))
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
MARK(U61(X)) → U611(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → ACTIVE(U71(mark(X)))
MARK(U71(X)) → U711(mark(X))
MARK(U71(X)) → MARK(X)
MARK(U81(X)) → ACTIVE(U81(mark(X)))
MARK(U81(X)) → U811(mark(X))
MARK(U81(X)) → MARK(X)
MARK(U91(X)) → ACTIVE(U91(mark(X)))
MARK(U91(X)) → U911(mark(X))
MARK(U91(X)) → MARK(X)
MARK(isPLNat(X)) → ACTIVE(isPLNat(X))
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)
MARK(0) → ACTIVE(0)
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)
U1011(mark(X1), X2) → U1011(X1, X2)
U1011(X1, mark(X2)) → U1011(X1, X2)
U1011(active(X1), X2) → U1011(X1, X2)
U1011(X1, active(X2)) → U1011(X1, X2)
U1021(mark(X)) → U1021(X)
U1021(active(X)) → U1021(X)
ISLNAT(mark(X)) → ISLNAT(X)
ISLNAT(active(X)) → ISLNAT(X)
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)
U1111(mark(X)) → U1111(X)
U1111(active(X)) → U1111(X)
SND(mark(X)) → SND(X)
SND(active(X)) → SND(X)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)
U1211(mark(X)) → U1211(X)
U1211(active(X)) → U1211(X)
U1311(mark(X1), X2) → U1311(X1, X2)
U1311(X1, mark(X2)) → U1311(X1, X2)
U1311(active(X1), X2) → U1311(X1, X2)
U1311(X1, active(X2)) → U1311(X1, X2)
U1321(mark(X)) → U1321(X)
U1321(active(X)) → U1321(X)
U1411(mark(X1), X2) → U1411(X1, X2)
U1411(X1, mark(X2)) → U1411(X1, X2)
U1411(active(X1), X2) → U1411(X1, X2)
U1411(X1, active(X2)) → U1411(X1, X2)
U1421(mark(X)) → U1421(X)
U1421(active(X)) → U1421(X)
U1511(mark(X1), X2) → U1511(X1, X2)
U1511(X1, mark(X2)) → U1511(X1, X2)
U1511(active(X1), X2) → U1511(X1, X2)
U1511(X1, active(X2)) → U1511(X1, X2)
U1521(mark(X)) → U1521(X)
U1521(active(X)) → U1521(X)
U1611(mark(X1), X2) → U1611(X1, X2)
U1611(X1, mark(X2)) → U1611(X1, X2)
U1611(active(X1), X2) → U1611(X1, X2)
U1611(X1, active(X2)) → U1611(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
NATSFROM(mark(X)) → NATSFROM(X)
NATSFROM(active(X)) → NATSFROM(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
U1711(mark(X1), X2, X3) → U1711(X1, X2, X3)
U1711(X1, mark(X2), X3) → U1711(X1, X2, X3)
U1711(X1, X2, mark(X3)) → U1711(X1, X2, X3)
U1711(active(X1), X2, X3) → U1711(X1, X2, X3)
U1711(X1, active(X2), X3) → U1711(X1, X2, X3)
U1711(X1, X2, active(X3)) → U1711(X1, X2, X3)
U1721(mark(X1), X2, X3) → U1721(X1, X2, X3)
U1721(X1, mark(X2), X3) → U1721(X1, X2, X3)
U1721(X1, X2, mark(X3)) → U1721(X1, X2, X3)
U1721(active(X1), X2, X3) → U1721(X1, X2, X3)
U1721(X1, active(X2), X3) → U1721(X1, X2, X3)
U1721(X1, X2, active(X3)) → U1721(X1, X2, X3)
HEAD(mark(X)) → HEAD(X)
HEAD(active(X)) → HEAD(X)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)
U1811(mark(X1), X2) → U1811(X1, X2)
U1811(X1, mark(X2)) → U1811(X1, X2)
U1811(active(X1), X2) → U1811(X1, X2)
U1811(X1, active(X2)) → U1811(X1, X2)
U1821(mark(X1), X2) → U1821(X1, X2)
U1821(X1, mark(X2)) → U1821(X1, X2)
U1821(active(X1), X2) → U1821(X1, X2)
U1821(X1, active(X2)) → U1821(X1, X2)
U1911(mark(X1), X2) → U1911(X1, X2)
U1911(X1, mark(X2)) → U1911(X1, X2)
U1911(active(X1), X2) → U1911(X1, X2)
U1911(X1, active(X2)) → U1911(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)
U2011(mark(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, mark(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, mark(X3), X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, X3, mark(X4)) → U2011(X1, X2, X3, X4)
U2011(active(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, active(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, active(X3), X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, X3, active(X4)) → U2011(X1, X2, X3, X4)
U2021(mark(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, mark(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, mark(X3), X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, X3, mark(X4)) → U2021(X1, X2, X3, X4)
U2021(active(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, active(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, active(X3), X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, X3, active(X4)) → U2021(X1, X2, X3, X4)
ISNATURAL(mark(X)) → ISNATURAL(X)
ISNATURAL(active(X)) → ISNATURAL(X)
U2031(mark(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, mark(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, mark(X3), X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, X3, mark(X4)) → U2031(X1, X2, X3, X4)
U2031(active(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, active(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, active(X3), X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, X3, active(X4)) → U2031(X1, X2, X3, X4)
U2041(mark(X1), X2) → U2041(X1, X2)
U2041(X1, mark(X2)) → U2041(X1, X2)
U2041(active(X1), X2) → U2041(X1, X2)
U2041(X1, active(X2)) → U2041(X1, X2)
U211(mark(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, mark(X2), X3) → U211(X1, X2, X3)
U211(X1, X2, mark(X3)) → U211(X1, X2, X3)
U211(active(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, active(X2), X3) → U211(X1, X2, X3)
U211(X1, X2, active(X3)) → U211(X1, X2, X3)
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)
U2111(mark(X1), X2) → U2111(X1, X2)
U2111(X1, mark(X2)) → U2111(X1, X2)
U2111(active(X1), X2) → U2111(X1, X2)
U2111(X1, active(X2)) → U2111(X1, X2)
U2121(mark(X1), X2) → U2121(X1, X2)
U2121(X1, mark(X2)) → U2121(X1, X2)
U2121(active(X1), X2) → U2121(X1, X2)
U2121(X1, active(X2)) → U2121(X1, X2)
U2211(mark(X1), X2, X3) → U2211(X1, X2, X3)
U2211(X1, mark(X2), X3) → U2211(X1, X2, X3)
U2211(X1, X2, mark(X3)) → U2211(X1, X2, X3)
U2211(active(X1), X2, X3) → U2211(X1, X2, X3)
U2211(X1, active(X2), X3) → U2211(X1, X2, X3)
U2211(X1, X2, active(X3)) → U2211(X1, X2, X3)
U2221(mark(X1), X2, X3) → U2221(X1, X2, X3)
U2221(X1, mark(X2), X3) → U2221(X1, X2, X3)
U2221(X1, X2, mark(X3)) → U2221(X1, X2, X3)
U2221(active(X1), X2, X3) → U2221(X1, X2, X3)
U2221(X1, active(X2), X3) → U2221(X1, X2, X3)
U2221(X1, X2, active(X3)) → U2221(X1, X2, X3)
FST(mark(X)) → FST(X)
FST(active(X)) → FST(X)
U311(mark(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, mark(X2), X3) → U311(X1, X2, X3)
U311(X1, X2, mark(X3)) → U311(X1, X2, X3)
U311(active(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, active(X2), X3) → U311(X1, X2, X3)
U311(X1, X2, active(X3)) → U311(X1, X2, X3)
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) → U411(X1, X2)
U411(X1, mark(X2)) → U411(X1, X2)
U411(active(X1), X2) → U411(X1, X2)
U411(X1, active(X2)) → U411(X1, X2)
U421(mark(X)) → U421(X)
U421(active(X)) → U421(X)
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(X)) → U521(X)
U521(active(X)) → U521(X)
U611(mark(X)) → U611(X)
U611(active(X)) → U611(X)
U711(mark(X)) → U711(X)
U711(active(X)) → U711(X)
U811(mark(X)) → U811(X)
U811(active(X)) → U811(X)
U911(mark(X)) → U911(X)
U911(active(X)) → U911(X)
ISPLNAT(mark(X)) → ISPLNAT(X)
ISPLNAT(active(X)) → ISPLNAT(X)
TAIL(mark(X)) → TAIL(X)
TAIL(active(X)) → TAIL(X)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 54 SCCs with 155 less nodes.

(4) Complex Obligation (AND)

(5) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


SEL(X1, mark(X2)) → SEL(X1, X2)
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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(9) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
active1 > SEL1

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


TAKE(X1, mark(X2)) → TAKE(X1, X2)
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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(18) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(19) 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: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > TAKE1

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(21) PisEmptyProof (EQUIVALENT transformation)

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

(22) TRUE

(23) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(25) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


TAIL(mark(X)) → TAIL(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > TAIL1

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

ISPLNAT(active(X)) → ISPLNAT(X)
ISPLNAT(mark(X)) → ISPLNAT(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

ISPLNAT(mark(X)) → ISPLNAT(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


ISPLNAT(mark(X)) → ISPLNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > ISPLNAT1

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

U911(active(X)) → U911(X)
U911(mark(X)) → U911(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U911(active(X)) → U911(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U911(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

U911(mark(X)) → U911(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U911(mark(X)) → U911(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U91^11

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

U811(active(X)) → U811(X)
U811(mark(X)) → U811(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

U811(mark(X)) → U811(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U811(mark(X)) → U811(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U81^11

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

U711(active(X)) → U711(X)
U711(mark(X)) → U711(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

U711(mark(X)) → U711(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U711(mark(X)) → U711(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U71^11

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(56) PisEmptyProof (EQUIVALENT transformation)

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

(57) TRUE

(58) Obligation:

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

U611(active(X)) → U611(X)
U611(mark(X)) → U611(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

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

U611(mark(X)) → U611(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(mark(X)) → U611(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U61^11

The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(63) PisEmptyProof (EQUIVALENT transformation)

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

(64) TRUE

(65) Obligation:

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

U521(active(X)) → U521(X)
U521(mark(X)) → U521(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(67) Obligation:

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

U521(mark(X)) → U521(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U521(mark(X)) → U521(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U52^11

The following usable rules [FROCOS05] were oriented: none

(69) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(70) PisEmptyProof (EQUIVALENT transformation)

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

(71) TRUE

(72) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U511(X1, mark(X2)) → U511(X1, X2)
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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U511(mark(X1), X2) → U511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U511(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(76) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(77) 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: Combined order from the following AFS and order.
U511(x1, x2)  =  U511(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U51^11

The following usable rules [FROCOS05] were oriented: none

(78) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(79) PisEmptyProof (EQUIVALENT transformation)

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

(80) TRUE

(81) Obligation:

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

U421(active(X)) → U421(X)
U421(mark(X)) → U421(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

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

U421(mark(X)) → U421(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(84) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U421(mark(X)) → U421(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U42^11

The following usable rules [FROCOS05] were oriented: none

(85) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(86) PisEmptyProof (EQUIVALENT transformation)

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

(87) TRUE

(88) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U411(X1, mark(X2)) → U411(X1, X2)
U411(X1, active(X2)) → U411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(90) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U411(mark(X1), X2) → U411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(92) Obligation:

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

U411(active(X1), X2) → U411(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U411(active(X1), X2) → U411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U411(x1, x2)  =  U411(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U41^11

The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(95) PisEmptyProof (EQUIVALENT transformation)

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

(96) TRUE

(97) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U321(X1, mark(X2)) → U321(X1, X2)
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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(99) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(101) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(102) 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: Combined order from the following AFS and order.
U321(x1, x2)  =  U321(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U32^11

The following usable rules [FROCOS05] were oriented: none

(103) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(104) PisEmptyProof (EQUIVALENT transformation)

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

(105) TRUE

(106) Obligation:

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

U311(X1, mark(X2), X3) → U311(X1, X2, X3)
U311(mark(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, X2, mark(X3)) → U311(X1, X2, X3)
U311(active(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, active(X2), X3) → U311(X1, X2, X3)
U311(X1, X2, active(X3)) → U311(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U311(X1, X2, mark(X3)) → U311(X1, X2, X3)
U311(X1, X2, active(X3)) → U311(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(108) Obligation:

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

U311(X1, mark(X2), X3) → U311(X1, X2, X3)
U311(mark(X1), X2, X3) → U311(X1, X2, X3)
U311(active(X1), X2, X3) → U311(X1, X2, X3)
U311(X1, active(X2), X3) → U311(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U311(mark(X1), X2, X3) → U311(X1, X2, X3)
U311(active(X1), X2, X3) → U311(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(110) Obligation:

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

U311(X1, mark(X2), X3) → U311(X1, X2, X3)
U311(X1, active(X2), X3) → U311(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U311(X1, mark(X2), X3) → U311(X1, X2, X3)
U311(X1, active(X2), X3) → U311(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U311(x1, x2, x3)  =  U311(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U31^11
active1 > U31^11

The following usable rules [FROCOS05] were oriented: none

(112) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(113) PisEmptyProof (EQUIVALENT transformation)

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

(114) TRUE

(115) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(117) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


FST(mark(X)) → FST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > FST1

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

U2221(X1, mark(X2), X3) → U2221(X1, X2, X3)
U2221(mark(X1), X2, X3) → U2221(X1, X2, X3)
U2221(X1, X2, mark(X3)) → U2221(X1, X2, X3)
U2221(active(X1), X2, X3) → U2221(X1, X2, X3)
U2221(X1, active(X2), X3) → U2221(X1, X2, X3)
U2221(X1, X2, active(X3)) → U2221(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2221(X1, X2, mark(X3)) → U2221(X1, X2, X3)
U2221(X1, X2, active(X3)) → U2221(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2221(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(124) Obligation:

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

U2221(X1, mark(X2), X3) → U2221(X1, X2, X3)
U2221(mark(X1), X2, X3) → U2221(X1, X2, X3)
U2221(active(X1), X2, X3) → U2221(X1, X2, X3)
U2221(X1, active(X2), X3) → U2221(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2221(mark(X1), X2, X3) → U2221(X1, X2, X3)
U2221(active(X1), X2, X3) → U2221(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2221(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(126) Obligation:

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

U2221(X1, mark(X2), X3) → U2221(X1, X2, X3)
U2221(X1, active(X2), X3) → U2221(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2221(X1, mark(X2), X3) → U2221(X1, X2, X3)
U2221(X1, active(X2), X3) → U2221(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2221(x1, x2, x3)  =  U2221(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U222^11
active1 > U222^11

The following usable rules [FROCOS05] were oriented: none

(128) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(129) PisEmptyProof (EQUIVALENT transformation)

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

(130) TRUE

(131) Obligation:

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

U2211(X1, mark(X2), X3) → U2211(X1, X2, X3)
U2211(mark(X1), X2, X3) → U2211(X1, X2, X3)
U2211(X1, X2, mark(X3)) → U2211(X1, X2, X3)
U2211(active(X1), X2, X3) → U2211(X1, X2, X3)
U2211(X1, active(X2), X3) → U2211(X1, X2, X3)
U2211(X1, X2, active(X3)) → U2211(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(132) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2211(X1, X2, mark(X3)) → U2211(X1, X2, X3)
U2211(X1, X2, active(X3)) → U2211(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2211(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(133) Obligation:

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

U2211(X1, mark(X2), X3) → U2211(X1, X2, X3)
U2211(mark(X1), X2, X3) → U2211(X1, X2, X3)
U2211(active(X1), X2, X3) → U2211(X1, X2, X3)
U2211(X1, active(X2), X3) → U2211(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(134) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2211(mark(X1), X2, X3) → U2211(X1, X2, X3)
U2211(active(X1), X2, X3) → U2211(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2211(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(135) Obligation:

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

U2211(X1, mark(X2), X3) → U2211(X1, X2, X3)
U2211(X1, active(X2), X3) → U2211(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2211(X1, mark(X2), X3) → U2211(X1, X2, X3)
U2211(X1, active(X2), X3) → U2211(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2211(x1, x2, x3)  =  U2211(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U221^11
active1 > U221^11

The following usable rules [FROCOS05] were oriented: none

(137) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(138) PisEmptyProof (EQUIVALENT transformation)

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

(139) TRUE

(140) Obligation:

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

U2121(X1, mark(X2)) → U2121(X1, X2)
U2121(mark(X1), X2) → U2121(X1, X2)
U2121(active(X1), X2) → U2121(X1, X2)
U2121(X1, active(X2)) → U2121(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(141) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2121(X1, mark(X2)) → U2121(X1, X2)
U2121(X1, active(X2)) → U2121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2121(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(142) Obligation:

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

U2121(mark(X1), X2) → U2121(X1, X2)
U2121(active(X1), X2) → U2121(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(143) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2121(mark(X1), X2) → U2121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2121(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(144) Obligation:

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

U2121(active(X1), X2) → U2121(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(145) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2121(active(X1), X2) → U2121(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2121(x1, x2)  =  U2121(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U212^11

The following usable rules [FROCOS05] were oriented: none

(146) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(147) PisEmptyProof (EQUIVALENT transformation)

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

(148) TRUE

(149) Obligation:

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

U2111(X1, mark(X2)) → U2111(X1, X2)
U2111(mark(X1), X2) → U2111(X1, X2)
U2111(active(X1), X2) → U2111(X1, X2)
U2111(X1, active(X2)) → U2111(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2111(X1, mark(X2)) → U2111(X1, X2)
U2111(X1, active(X2)) → U2111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2111(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(151) Obligation:

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

U2111(mark(X1), X2) → U2111(X1, X2)
U2111(active(X1), X2) → U2111(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2111(mark(X1), X2) → U2111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2111(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(153) Obligation:

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

U2111(active(X1), X2) → U2111(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(154) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2111(active(X1), X2) → U2111(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2111(x1, x2)  =  U2111(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U211^11

The following usable rules [FROCOS05] were oriented: none

(155) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(156) PisEmptyProof (EQUIVALENT transformation)

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

(157) TRUE

(158) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U221(X1, mark(X2)) → U221(X1, X2)
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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(160) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(162) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U221(active(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)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U22^11

The following usable rules [FROCOS05] were oriented: none

(164) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(165) PisEmptyProof (EQUIVALENT transformation)

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

(166) TRUE

(167) Obligation:

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

U211(X1, mark(X2), X3) → U211(X1, X2, X3)
U211(mark(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, X2, mark(X3)) → U211(X1, X2, X3)
U211(active(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, active(X2), X3) → U211(X1, X2, X3)
U211(X1, X2, active(X3)) → U211(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(168) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(X1, X2, mark(X3)) → U211(X1, X2, X3)
U211(X1, X2, active(X3)) → U211(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(169) Obligation:

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

U211(X1, mark(X2), X3) → U211(X1, X2, X3)
U211(mark(X1), X2, X3) → U211(X1, X2, X3)
U211(active(X1), X2, X3) → U211(X1, X2, X3)
U211(X1, active(X2), X3) → U211(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(170) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(mark(X1), X2, X3) → U211(X1, X2, X3)
U211(active(X1), X2, X3) → U211(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(171) Obligation:

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

U211(X1, mark(X2), X3) → U211(X1, X2, X3)
U211(X1, active(X2), X3) → U211(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(172) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(X1, mark(X2), X3) → U211(X1, X2, X3)
U211(X1, active(X2), X3) → U211(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1, x2, x3)  =  U211(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U21^11
active1 > U21^11

The following usable rules [FROCOS05] were oriented: none

(173) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(174) PisEmptyProof (EQUIVALENT transformation)

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

(175) TRUE

(176) Obligation:

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

U2041(X1, mark(X2)) → U2041(X1, X2)
U2041(mark(X1), X2) → U2041(X1, X2)
U2041(active(X1), X2) → U2041(X1, X2)
U2041(X1, active(X2)) → U2041(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(177) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2041(X1, mark(X2)) → U2041(X1, X2)
U2041(X1, active(X2)) → U2041(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2041(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(178) Obligation:

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

U2041(mark(X1), X2) → U2041(X1, X2)
U2041(active(X1), X2) → U2041(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(179) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2041(mark(X1), X2) → U2041(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2041(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(180) Obligation:

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

U2041(active(X1), X2) → U2041(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(181) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2041(active(X1), X2) → U2041(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2041(x1, x2)  =  U2041(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U204^11

The following usable rules [FROCOS05] were oriented: none

(182) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(183) PisEmptyProof (EQUIVALENT transformation)

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

(184) TRUE

(185) Obligation:

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

U2031(X1, mark(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(mark(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, mark(X3), X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, X3, mark(X4)) → U2031(X1, X2, X3, X4)
U2031(active(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, active(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, active(X3), X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, X3, active(X4)) → U2031(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(186) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2031(X1, X2, mark(X3), X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, active(X3), X4) → U2031(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2031(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(187) Obligation:

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

U2031(X1, mark(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(mark(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, X3, mark(X4)) → U2031(X1, X2, X3, X4)
U2031(active(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, active(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, X2, X3, active(X4)) → U2031(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(188) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2031(X1, X2, X3, mark(X4)) → U2031(X1, X2, X3, X4)
U2031(X1, X2, X3, active(X4)) → U2031(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2031(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(189) Obligation:

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

U2031(X1, mark(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(mark(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(active(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(X1, active(X2), X3, X4) → U2031(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(190) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2031(X1, active(X2), X3, X4) → U2031(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2031(x1, x2, x3, x4)  =  U2031(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(191) Obligation:

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

U2031(X1, mark(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(mark(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
U2031(active(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2031(mark(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2031(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(193) Obligation:

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

U2031(X1, mark(X2), X3, X4) → U2031(X1, X2, X3, X4)
U2031(active(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2031(X1, mark(X2), X3, X4) → U2031(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2031(x1, x2, x3, x4)  =  U2031(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U203^11
active1 > U203^11

The following usable rules [FROCOS05] were oriented: none

(195) Obligation:

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

U2031(active(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2031(active(X1), X2, X3, X4) → U2031(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2031(x1, x2, x3, x4)  =  U2031(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

ISNATURAL(active(X)) → ISNATURAL(X)
ISNATURAL(mark(X)) → ISNATURAL(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(202) Obligation:

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

ISNATURAL(mark(X)) → ISNATURAL(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


ISNATURAL(mark(X)) → ISNATURAL(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > ISNATURAL1

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

U2021(X1, mark(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(mark(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, mark(X3), X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, X3, mark(X4)) → U2021(X1, X2, X3, X4)
U2021(active(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, active(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, active(X3), X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, X3, active(X4)) → U2021(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2021(X1, X2, mark(X3), X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, active(X3), X4) → U2021(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2021(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(209) Obligation:

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

U2021(X1, mark(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(mark(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, X3, mark(X4)) → U2021(X1, X2, X3, X4)
U2021(active(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, active(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, X2, X3, active(X4)) → U2021(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2021(X1, X2, X3, mark(X4)) → U2021(X1, X2, X3, X4)
U2021(X1, X2, X3, active(X4)) → U2021(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2021(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(211) Obligation:

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

U2021(X1, mark(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(mark(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(active(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(X1, active(X2), X3, X4) → U2021(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2021(X1, active(X2), X3, X4) → U2021(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2021(x1, x2, x3, x4)  =  U2021(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(213) Obligation:

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

U2021(X1, mark(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(mark(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
U2021(active(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2021(mark(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2021(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(215) Obligation:

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

U2021(X1, mark(X2), X3, X4) → U2021(X1, X2, X3, X4)
U2021(active(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2021(X1, mark(X2), X3, X4) → U2021(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2021(x1, x2, x3, x4)  =  U2021(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U202^11
active1 > U202^11

The following usable rules [FROCOS05] were oriented: none

(217) Obligation:

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

U2021(active(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(218) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2021(active(X1), X2, X3, X4) → U2021(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2021(x1, x2, x3, x4)  =  U2021(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(219) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(220) PisEmptyProof (EQUIVALENT transformation)

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

(221) TRUE

(222) Obligation:

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

U2011(X1, mark(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(mark(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, mark(X3), X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, X3, mark(X4)) → U2011(X1, X2, X3, X4)
U2011(active(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, active(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, active(X3), X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, X3, active(X4)) → U2011(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2011(X1, X2, mark(X3), X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, active(X3), X4) → U2011(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2011(x1, x2, x3, x4)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(224) Obligation:

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

U2011(X1, mark(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(mark(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, X3, mark(X4)) → U2011(X1, X2, X3, X4)
U2011(active(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, active(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, X2, X3, active(X4)) → U2011(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2011(X1, X2, X3, mark(X4)) → U2011(X1, X2, X3, X4)
U2011(X1, X2, X3, active(X4)) → U2011(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2011(x1, x2, x3, x4)  =  x4
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(226) Obligation:

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

U2011(X1, mark(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(mark(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(active(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(X1, active(X2), X3, X4) → U2011(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2011(X1, active(X2), X3, X4) → U2011(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2011(x1, x2, x3, x4)  =  U2011(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(228) Obligation:

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

U2011(X1, mark(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(mark(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
U2011(active(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U2011(mark(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2011(x1, x2, x3, x4)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(230) Obligation:

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

U2011(X1, mark(X2), X3, X4) → U2011(X1, X2, X3, X4)
U2011(active(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(231) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2011(X1, mark(X2), X3, X4) → U2011(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2011(x1, x2, x3, x4)  =  U2011(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U201^11
active1 > U201^11

The following usable rules [FROCOS05] were oriented: none

(232) Obligation:

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

U2011(active(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(233) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U2011(active(X1), X2, X3, X4) → U2011(X1, X2, X3, X4)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U2011(x1, x2, x3, x4)  =  U2011(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(234) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(235) PisEmptyProof (EQUIVALENT transformation)

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

(236) TRUE

(237) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


PAIR(X1, mark(X2)) → PAIR(X1, X2)
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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(239) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(240) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(241) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(242) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
active1 > PAIR1

The following usable rules [FROCOS05] were oriented: none

(243) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(244) PisEmptyProof (EQUIVALENT transformation)

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

(245) TRUE

(246) Obligation:

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

U1911(X1, mark(X2)) → U1911(X1, X2)
U1911(mark(X1), X2) → U1911(X1, X2)
U1911(active(X1), X2) → U1911(X1, X2)
U1911(X1, active(X2)) → U1911(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1911(X1, mark(X2)) → U1911(X1, X2)
U1911(X1, active(X2)) → U1911(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1911(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(248) Obligation:

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

U1911(mark(X1), X2) → U1911(X1, X2)
U1911(active(X1), X2) → U1911(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(249) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1911(mark(X1), X2) → U1911(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1911(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(250) Obligation:

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

U1911(active(X1), X2) → U1911(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(251) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1911(active(X1), X2) → U1911(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1911(x1, x2)  =  U1911(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U191^11

The following usable rules [FROCOS05] were oriented: none

(252) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(253) PisEmptyProof (EQUIVALENT transformation)

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

(254) TRUE

(255) Obligation:

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

U1821(X1, mark(X2)) → U1821(X1, X2)
U1821(mark(X1), X2) → U1821(X1, X2)
U1821(active(X1), X2) → U1821(X1, X2)
U1821(X1, active(X2)) → U1821(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1821(X1, mark(X2)) → U1821(X1, X2)
U1821(X1, active(X2)) → U1821(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1821(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(257) Obligation:

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

U1821(mark(X1), X2) → U1821(X1, X2)
U1821(active(X1), X2) → U1821(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(258) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1821(mark(X1), X2) → U1821(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1821(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(259) Obligation:

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

U1821(active(X1), X2) → U1821(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(260) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1821(active(X1), X2) → U1821(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1821(x1, x2)  =  U1821(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U182^11

The following usable rules [FROCOS05] were oriented: none

(261) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(262) PisEmptyProof (EQUIVALENT transformation)

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

(263) TRUE

(264) Obligation:

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

U1811(X1, mark(X2)) → U1811(X1, X2)
U1811(mark(X1), X2) → U1811(X1, X2)
U1811(active(X1), X2) → U1811(X1, X2)
U1811(X1, active(X2)) → U1811(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1811(X1, mark(X2)) → U1811(X1, X2)
U1811(X1, active(X2)) → U1811(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1811(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(266) Obligation:

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

U1811(mark(X1), X2) → U1811(X1, X2)
U1811(active(X1), X2) → U1811(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(267) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1811(mark(X1), X2) → U1811(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1811(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(268) Obligation:

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

U1811(active(X1), X2) → U1811(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(269) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1811(active(X1), X2) → U1811(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1811(x1, x2)  =  U1811(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U181^11

The following usable rules [FROCOS05] were oriented: none

(270) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(271) PisEmptyProof (EQUIVALENT transformation)

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

(272) TRUE

(273) Obligation:

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

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(275) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(276) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(277) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(278) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
active1 > AFTERNTH1

The following usable rules [FROCOS05] were oriented: none

(279) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(280) PisEmptyProof (EQUIVALENT transformation)

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

(281) TRUE

(282) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(283) 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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(284) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(285) 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: Lexicographic Path Order [LPO].
Precedence:
mark1 > HEAD1

The following usable rules [FROCOS05] were oriented: none

(286) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(287) PisEmptyProof (EQUIVALENT transformation)

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

(288) TRUE

(289) Obligation:

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

U1721(X1, mark(X2), X3) → U1721(X1, X2, X3)
U1721(mark(X1), X2, X3) → U1721(X1, X2, X3)
U1721(X1, X2, mark(X3)) → U1721(X1, X2, X3)
U1721(active(X1), X2, X3) → U1721(X1, X2, X3)
U1721(X1, active(X2), X3) → U1721(X1, X2, X3)
U1721(X1, X2, active(X3)) → U1721(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1721(X1, X2, mark(X3)) → U1721(X1, X2, X3)
U1721(X1, X2, active(X3)) → U1721(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1721(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(291) Obligation:

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

U1721(X1, mark(X2), X3) → U1721(X1, X2, X3)
U1721(mark(X1), X2, X3) → U1721(X1, X2, X3)
U1721(active(X1), X2, X3) → U1721(X1, X2, X3)
U1721(X1, active(X2), X3) → U1721(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1721(mark(X1), X2, X3) → U1721(X1, X2, X3)
U1721(active(X1), X2, X3) → U1721(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1721(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(293) Obligation:

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

U1721(X1, mark(X2), X3) → U1721(X1, X2, X3)
U1721(X1, active(X2), X3) → U1721(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1721(X1, mark(X2), X3) → U1721(X1, X2, X3)
U1721(X1, active(X2), X3) → U1721(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1721(x1, x2, x3)  =  U1721(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U172^11
active1 > U172^11

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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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:

U1711(X1, mark(X2), X3) → U1711(X1, X2, X3)
U1711(mark(X1), X2, X3) → U1711(X1, X2, X3)
U1711(X1, X2, mark(X3)) → U1711(X1, X2, X3)
U1711(active(X1), X2, X3) → U1711(X1, X2, X3)
U1711(X1, active(X2), X3) → U1711(X1, X2, X3)
U1711(X1, X2, active(X3)) → U1711(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1711(X1, X2, mark(X3)) → U1711(X1, X2, X3)
U1711(X1, X2, active(X3)) → U1711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1711(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(300) Obligation:

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

U1711(X1, mark(X2), X3) → U1711(X1, X2, X3)
U1711(mark(X1), X2, X3) → U1711(X1, X2, X3)
U1711(active(X1), X2, X3) → U1711(X1, X2, X3)
U1711(X1, active(X2), X3) → U1711(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1711(mark(X1), X2, X3) → U1711(X1, X2, X3)
U1711(active(X1), X2, X3) → U1711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1711(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(302) Obligation:

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

U1711(X1, mark(X2), X3) → U1711(X1, X2, X3)
U1711(X1, active(X2), X3) → U1711(X1, X2, X3)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(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.


U1711(X1, mark(X2), X3) → U1711(X1, X2, X3)
U1711(X1, active(X2), X3) → U1711(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1711(x1, x2, x3)  =  U1711(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U171^11
active1 > U171^11

The following usable rules [FROCOS05] were oriented: none

(304) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(305) PisEmptyProof (EQUIVALENT transformation)

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

(306) TRUE

(307) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(308) 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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(309) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(310) 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: Lexicographic Path Order [LPO].
Precedence:
mark1 > S1

The following usable rules [FROCOS05] were oriented: none

(311) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(312) PisEmptyProof (EQUIVALENT transformation)

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

(313) TRUE

(314) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(315) 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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(316) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(317) 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: Lexicographic Path Order [LPO].
Precedence:
mark1 > NATSFROM1

The following usable rules [FROCOS05] were oriented: none

(318) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(319) PisEmptyProof (EQUIVALENT transformation)

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

(320) TRUE

(321) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(322) 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(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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(323) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(324) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(mark(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(325) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(326) 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: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > CONS1

The following usable rules [FROCOS05] were oriented: none

(327) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(328) PisEmptyProof (EQUIVALENT transformation)

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

(329) TRUE

(330) Obligation:

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

U1611(X1, mark(X2)) → U1611(X1, X2)
U1611(mark(X1), X2) → U1611(X1, X2)
U1611(active(X1), X2) → U1611(X1, X2)
U1611(X1, active(X2)) → U1611(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(331) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1611(X1, mark(X2)) → U1611(X1, X2)
U1611(X1, active(X2)) → U1611(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1611(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(332) Obligation:

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

U1611(mark(X1), X2) → U1611(X1, X2)
U1611(active(X1), X2) → U1611(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(333) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1611(mark(X1), X2) → U1611(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1611(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(334) Obligation:

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

U1611(active(X1), X2) → U1611(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(335) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1611(active(X1), X2) → U1611(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1611(x1, x2)  =  U1611(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U161^11

The following usable rules [FROCOS05] were oriented: none

(336) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(337) PisEmptyProof (EQUIVALENT transformation)

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

(338) TRUE

(339) Obligation:

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

U1521(active(X)) → U1521(X)
U1521(mark(X)) → U1521(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(340) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1521(active(X)) → U1521(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1521(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(341) Obligation:

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

U1521(mark(X)) → U1521(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(342) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1521(mark(X)) → U1521(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U152^11

The following usable rules [FROCOS05] were oriented: none

(343) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(344) PisEmptyProof (EQUIVALENT transformation)

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

(345) TRUE

(346) Obligation:

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

U1511(X1, mark(X2)) → U1511(X1, X2)
U1511(mark(X1), X2) → U1511(X1, X2)
U1511(active(X1), X2) → U1511(X1, X2)
U1511(X1, active(X2)) → U1511(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(347) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1511(X1, mark(X2)) → U1511(X1, X2)
U1511(X1, active(X2)) → U1511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1511(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(348) Obligation:

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

U1511(mark(X1), X2) → U1511(X1, X2)
U1511(active(X1), X2) → U1511(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(349) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1511(mark(X1), X2) → U1511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1511(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(350) Obligation:

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

U1511(active(X1), X2) → U1511(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(351) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1511(active(X1), X2) → U1511(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1511(x1, x2)  =  U1511(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U151^11

The following usable rules [FROCOS05] were oriented: none

(352) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(353) PisEmptyProof (EQUIVALENT transformation)

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

(354) TRUE

(355) Obligation:

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

U1421(active(X)) → U1421(X)
U1421(mark(X)) → U1421(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(356) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1421(active(X)) → U1421(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1421(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(357) Obligation:

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

U1421(mark(X)) → U1421(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(358) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1421(mark(X)) → U1421(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U142^11

The following usable rules [FROCOS05] were oriented: none

(359) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(360) PisEmptyProof (EQUIVALENT transformation)

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

(361) TRUE

(362) Obligation:

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

U1411(X1, mark(X2)) → U1411(X1, X2)
U1411(mark(X1), X2) → U1411(X1, X2)
U1411(active(X1), X2) → U1411(X1, X2)
U1411(X1, active(X2)) → U1411(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(363) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1411(X1, mark(X2)) → U1411(X1, X2)
U1411(X1, active(X2)) → U1411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1411(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(364) Obligation:

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

U1411(mark(X1), X2) → U1411(X1, X2)
U1411(active(X1), X2) → U1411(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(365) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1411(mark(X1), X2) → U1411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1411(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(366) Obligation:

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

U1411(active(X1), X2) → U1411(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(367) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1411(active(X1), X2) → U1411(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1411(x1, x2)  =  U1411(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U141^11

The following usable rules [FROCOS05] were oriented: none

(368) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(369) PisEmptyProof (EQUIVALENT transformation)

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

(370) TRUE

(371) Obligation:

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

U1321(active(X)) → U1321(X)
U1321(mark(X)) → U1321(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(372) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1321(active(X)) → U1321(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1321(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(373) Obligation:

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

U1321(mark(X)) → U1321(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(374) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1321(mark(X)) → U1321(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U132^11

The following usable rules [FROCOS05] were oriented: none

(375) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(376) PisEmptyProof (EQUIVALENT transformation)

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

(377) TRUE

(378) Obligation:

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

U1311(X1, mark(X2)) → U1311(X1, X2)
U1311(mark(X1), X2) → U1311(X1, X2)
U1311(active(X1), X2) → U1311(X1, X2)
U1311(X1, active(X2)) → U1311(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(379) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1311(X1, mark(X2)) → U1311(X1, X2)
U1311(X1, active(X2)) → U1311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1311(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(380) Obligation:

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

U1311(mark(X1), X2) → U1311(X1, X2)
U1311(active(X1), X2) → U1311(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(381) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1311(mark(X1), X2) → U1311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1311(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(382) Obligation:

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

U1311(active(X1), X2) → U1311(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(383) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1311(active(X1), X2) → U1311(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1311(x1, x2)  =  U1311(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U131^11

The following usable rules [FROCOS05] were oriented: none

(384) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(385) PisEmptyProof (EQUIVALENT transformation)

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

(386) TRUE

(387) Obligation:

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

U1211(active(X)) → U1211(X)
U1211(mark(X)) → U1211(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(388) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1211(active(X)) → U1211(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1211(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(389) Obligation:

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

U1211(mark(X)) → U1211(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(390) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1211(mark(X)) → U1211(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U121^11

The following usable rules [FROCOS05] were oriented: none

(391) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(392) PisEmptyProof (EQUIVALENT transformation)

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

(393) TRUE

(394) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(395) 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(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)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(396) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(397) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(398) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(399) 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: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  SPLITAT(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > SPLITAT1

The following usable rules [FROCOS05] were oriented: none

(400) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(401) PisEmptyProof (EQUIVALENT transformation)

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

(402) TRUE

(403) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(404) 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)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(405) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(406) 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: Lexicographic Path Order [LPO].
Precedence:
mark1 > SND1

The following usable rules [FROCOS05] were oriented: none

(407) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(408) PisEmptyProof (EQUIVALENT transformation)

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

(409) TRUE

(410) Obligation:

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

U1111(active(X)) → U1111(X)
U1111(mark(X)) → U1111(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(411) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1111(active(X)) → U1111(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1111(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(412) Obligation:

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

U1111(mark(X)) → U1111(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(413) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1111(mark(X)) → U1111(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U111^11

The following usable rules [FROCOS05] were oriented: none

(414) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(415) PisEmptyProof (EQUIVALENT transformation)

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

(416) TRUE

(417) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(418) 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)
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
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(419) 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)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(420) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(mark(X1), X2, X3) → U121(X1, X2, X3)
U121(active(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)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(421) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(422) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U121(X1, mark(X2), X3) → U121(X1, X2, X3)
U121(X1, 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)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U12^11
active1 > U12^11

The following usable rules [FROCOS05] were oriented: none

(423) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(424) PisEmptyProof (EQUIVALENT transformation)

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

(425) TRUE

(426) 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(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(427) 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)
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
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(428) 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)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(429) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(mark(X1), X2, X3) → U111(X1, X2, X3)
U111(active(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)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(430) Obligation:

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

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

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(431) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U111(X1, mark(X2), X3) → U111(X1, X2, X3)
U111(X1, 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)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
mark1 > U11^11
active1 > U11^11

The following usable rules [FROCOS05] were oriented: none

(432) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(433) PisEmptyProof (EQUIVALENT transformation)

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

(434) TRUE

(435) Obligation:

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

ISLNAT(active(X)) → ISLNAT(X)
ISLNAT(mark(X)) → ISLNAT(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(436) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(437) Obligation:

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

ISLNAT(mark(X)) → ISLNAT(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(438) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ISLNAT(mark(X)) → ISLNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > ISLNAT1

The following usable rules [FROCOS05] were oriented: none

(439) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(440) PisEmptyProof (EQUIVALENT transformation)

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

(441) TRUE

(442) Obligation:

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

U1021(active(X)) → U1021(X)
U1021(mark(X)) → U1021(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(443) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1021(active(X)) → U1021(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1021(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(444) Obligation:

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

U1021(mark(X)) → U1021(X)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(445) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1021(mark(X)) → U1021(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
mark1 > U102^11

The following usable rules [FROCOS05] were oriented: none

(446) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(447) PisEmptyProof (EQUIVALENT transformation)

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

(448) TRUE

(449) Obligation:

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

U1011(X1, mark(X2)) → U1011(X1, X2)
U1011(mark(X1), X2) → U1011(X1, X2)
U1011(active(X1), X2) → U1011(X1, X2)
U1011(X1, active(X2)) → U1011(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(450) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1011(X1, mark(X2)) → U1011(X1, X2)
U1011(X1, active(X2)) → U1011(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1011(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(451) Obligation:

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

U1011(mark(X1), X2) → U1011(X1, X2)
U1011(active(X1), X2) → U1011(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(452) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1011(mark(X1), X2) → U1011(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1011(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:
trivial

The following usable rules [FROCOS05] were oriented: none

(453) Obligation:

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

U1011(active(X1), X2) → U1011(X1, X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(454) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U1011(active(X1), X2) → U1011(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U1011(x1, x2)  =  U1011(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:
active1 > U101^11

The following usable rules [FROCOS05] were oriented: none

(455) Obligation:

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

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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

(456) PisEmptyProof (EQUIVALENT transformation)

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

(457) TRUE

(458) Obligation:

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

MARK(U101(X1, X2)) → ACTIVE(U101(mark(X1), X2))
ACTIVE(U101(tt, V2)) → MARK(U102(isLNat(V2)))
MARK(U101(X1, X2)) → MARK(X1)
MARK(U102(X)) → ACTIVE(U102(mark(X)))
ACTIVE(U11(tt, N, XS)) → MARK(U12(isLNat(XS), N, XS))
MARK(U102(X)) → MARK(X)
MARK(isLNat(X)) → ACTIVE(isLNat(X))
ACTIVE(U12(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U131(tt, V2)) → MARK(U132(isLNat(V2)))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U141(tt, V2)) → MARK(U142(isLNat(V2)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(U111(X)) → ACTIVE(U111(mark(X)))
ACTIVE(U151(tt, V2)) → MARK(U152(isLNat(V2)))
MARK(U111(X)) → MARK(X)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(U161(tt, N)) → MARK(cons(N, natsFrom(s(N))))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(U171(tt, N, XS)) → MARK(U172(isLNat(XS), N, XS))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U121(X)) → ACTIVE(U121(mark(X)))
ACTIVE(U172(tt, N, XS)) → MARK(head(afterNth(N, XS)))
MARK(U121(X)) → MARK(X)
MARK(U131(X1, X2)) → ACTIVE(U131(mark(X1), X2))
ACTIVE(U181(tt, Y)) → MARK(U182(isLNat(Y), Y))
MARK(U131(X1, X2)) → MARK(X1)
MARK(U132(X)) → ACTIVE(U132(mark(X)))
ACTIVE(U182(tt, Y)) → MARK(Y)
MARK(U132(X)) → MARK(X)
MARK(U141(X1, X2)) → ACTIVE(U141(mark(X1), X2))
ACTIVE(U191(tt, XS)) → MARK(pair(nil, XS))
MARK(U141(X1, X2)) → MARK(X1)
MARK(U142(X)) → ACTIVE(U142(mark(X)))
ACTIVE(U201(tt, N, X, XS)) → MARK(U202(isNatural(X), N, X, XS))
MARK(U142(X)) → MARK(X)
MARK(U151(X1, X2)) → ACTIVE(U151(mark(X1), X2))
ACTIVE(U202(tt, N, X, XS)) → MARK(U203(isLNat(XS), N, X, XS))
MARK(U151(X1, X2)) → MARK(X1)
MARK(U152(X)) → ACTIVE(U152(mark(X)))
ACTIVE(U203(tt, N, X, XS)) → MARK(U204(splitAt(N, XS), X))
MARK(U152(X)) → MARK(X)
MARK(U161(X1, X2)) → ACTIVE(U161(mark(X1), X2))
ACTIVE(U204(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U161(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U21(tt, X, Y)) → MARK(U22(isLNat(Y), X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(U211(tt, XS)) → MARK(U212(isLNat(XS), XS))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(U212(tt, XS)) → MARK(XS)
MARK(s(X)) → MARK(X)
MARK(U171(X1, X2, X3)) → ACTIVE(U171(mark(X1), X2, X3))
ACTIVE(U22(tt, X)) → MARK(X)
MARK(U171(X1, X2, X3)) → MARK(X1)
MARK(U172(X1, X2, X3)) → ACTIVE(U172(mark(X1), X2, X3))
ACTIVE(U221(tt, N, XS)) → MARK(U222(isLNat(XS), N, XS))
MARK(U172(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(U222(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(U31(tt, N, XS)) → MARK(U32(isLNat(XS), N))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U181(X1, X2)) → ACTIVE(U181(mark(X1), X2))
ACTIVE(U32(tt, N)) → MARK(N)
MARK(U181(X1, X2)) → MARK(X1)
MARK(U182(X1, X2)) → ACTIVE(U182(mark(X1), X2))
ACTIVE(U41(tt, V2)) → MARK(U42(isLNat(V2)))
MARK(U182(X1, X2)) → MARK(X1)
MARK(U191(X1, X2)) → ACTIVE(U191(mark(X1), X2))
ACTIVE(U51(tt, V2)) → MARK(U52(isLNat(V2)))
MARK(U191(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
ACTIVE(afterNth(N, XS)) → MARK(U11(isNatural(N), N, XS))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(U201(X1, X2, X3, X4)) → ACTIVE(U201(mark(X1), X2, X3, X4))
ACTIVE(fst(pair(X, Y))) → MARK(U21(isLNat(X), X, Y))
MARK(U201(X1, X2, X3, X4)) → MARK(X1)
MARK(U202(X1, X2, X3, X4)) → ACTIVE(U202(mark(X1), X2, X3, X4))
ACTIVE(head(cons(N, XS))) → MARK(U31(isNatural(N), N, XS))
MARK(U202(X1, X2, X3, X4)) → MARK(X1)
MARK(isNatural(X)) → ACTIVE(isNatural(X))
ACTIVE(isLNat(afterNth(V1, V2))) → MARK(U41(isNatural(V1), V2))
MARK(U203(X1, X2, X3, X4)) → ACTIVE(U203(mark(X1), X2, X3, X4))
ACTIVE(isLNat(cons(V1, V2))) → MARK(U51(isNatural(V1), V2))
MARK(U203(X1, X2, X3, X4)) → MARK(X1)
MARK(U204(X1, X2)) → ACTIVE(U204(mark(X1), X2))
ACTIVE(isLNat(fst(V1))) → MARK(U61(isPLNat(V1)))
MARK(U204(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → ACTIVE(U21(mark(X1), X2, X3))
ACTIVE(isLNat(natsFrom(V1))) → MARK(U71(isNatural(V1)))
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(isLNat(snd(V1))) → MARK(U81(isPLNat(V1)))
MARK(U22(X1, X2)) → MARK(X1)
MARK(U211(X1, X2)) → ACTIVE(U211(mark(X1), X2))
ACTIVE(isLNat(tail(V1))) → MARK(U91(isLNat(V1)))
MARK(U211(X1, X2)) → MARK(X1)
MARK(U212(X1, X2)) → ACTIVE(U212(mark(X1), X2))
ACTIVE(isLNat(take(V1, V2))) → MARK(U101(isNatural(V1), V2))
MARK(U212(X1, X2)) → MARK(X1)
MARK(U221(X1, X2, X3)) → ACTIVE(U221(mark(X1), X2, X3))
ACTIVE(isNatural(head(V1))) → MARK(U111(isLNat(V1)))
MARK(U221(X1, X2, X3)) → MARK(X1)
MARK(U222(X1, X2, X3)) → ACTIVE(U222(mark(X1), X2, X3))
ACTIVE(isNatural(s(V1))) → MARK(U121(isNatural(V1)))
MARK(U222(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(isNatural(sel(V1, V2))) → MARK(U131(isNatural(V1), V2))
MARK(fst(X)) → MARK(X)
MARK(U31(X1, X2, X3)) → ACTIVE(U31(mark(X1), X2, X3))
ACTIVE(isPLNat(pair(V1, V2))) → MARK(U141(isLNat(V1), V2))
MARK(U31(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(isPLNat(splitAt(V1, V2))) → MARK(U151(isNatural(V1), V2))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → ACTIVE(U41(mark(X1), X2))
ACTIVE(natsFrom(N)) → MARK(U161(isNatural(N), N))
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → ACTIVE(U42(mark(X)))
ACTIVE(sel(N, XS)) → MARK(U171(isNatural(N), N, XS))
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(snd(pair(X, Y))) → MARK(U181(isLNat(X), Y))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(U191(isLNat(XS), XS))
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U201(isNatural(N), N, X, XS))
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → ACTIVE(U71(mark(X)))
ACTIVE(tail(cons(N, XS))) → MARK(U211(isNatural(N), XS))
MARK(U71(X)) → MARK(X)
MARK(U81(X)) → ACTIVE(U81(mark(X)))
ACTIVE(take(N, XS)) → MARK(U221(isNatural(N), N, XS))
MARK(U81(X)) → MARK(X)
MARK(U91(X)) → ACTIVE(U91(mark(X)))
MARK(U91(X)) → MARK(X)
MARK(isPLNat(X)) → ACTIVE(isPLNat(X))
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)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
mark(U101(X1, X2)) → active(U101(mark(X1), X2))
mark(tt) → active(tt)
mark(U102(X)) → active(U102(mark(X)))
mark(isLNat(X)) → active(isLNat(X))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(U111(X)) → active(U111(mark(X)))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U121(X)) → active(U121(mark(X)))
mark(U131(X1, X2)) → active(U131(mark(X1), X2))
mark(U132(X)) → active(U132(mark(X)))
mark(U141(X1, X2)) → active(U141(mark(X1), X2))
mark(U142(X)) → active(U142(mark(X)))
mark(U151(X1, X2)) → active(U151(mark(X1), X2))
mark(U152(X)) → active(U152(mark(X)))
mark(U161(X1, X2)) → active(U161(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(U171(X1, X2, X3)) → active(U171(mark(X1), X2, X3))
mark(U172(X1, X2, X3)) → active(U172(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U181(X1, X2)) → active(U181(mark(X1), X2))
mark(U182(X1, X2)) → active(U182(mark(X1), X2))
mark(U191(X1, X2)) → active(U191(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U201(X1, X2, X3, X4)) → active(U201(mark(X1), X2, X3, X4))
mark(U202(X1, X2, X3, X4)) → active(U202(mark(X1), X2, X3, X4))
mark(isNatural(X)) → active(isNatural(X))
mark(U203(X1, X2, X3, X4)) → active(U203(mark(X1), X2, X3, X4))
mark(U204(X1, X2)) → active(U204(mark(X1), X2))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U211(X1, X2)) → active(U211(mark(X1), X2))
mark(U212(X1, X2)) → active(U212(mark(X1), X2))
mark(U221(X1, X2, X3)) → active(U221(mark(X1), X2, X3))
mark(U222(X1, X2, X3)) → active(U222(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(U31(X1, X2, X3)) → active(U31(mark(X1), X2, X3))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U42(X)) → active(U42(mark(X)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X)) → active(U81(mark(X)))
mark(U91(X)) → active(U91(mark(X)))
mark(isPLNat(X)) → active(isPLNat(X))
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
mark(0) → active(0)
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
U101(mark(X1), X2) → U101(X1, X2)
U101(X1, mark(X2)) → U101(X1, X2)
U101(active(X1), X2) → U101(X1, X2)
U101(X1, active(X2)) → U101(X1, X2)
U102(mark(X)) → U102(X)
U102(active(X)) → U102(X)
isLNat(mark(X)) → isLNat(X)
isLNat(active(X)) → isLNat(X)
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
U111(mark(X)) → U111(X)
U111(active(X)) → U111(X)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
U131(mark(X1), X2) → U131(X1, X2)
U131(X1, mark(X2)) → U131(X1, X2)
U131(active(X1), X2) → U131(X1, X2)
U131(X1, active(X2)) → U131(X1, X2)
U132(mark(X)) → U132(X)
U132(active(X)) → U132(X)
U141(mark(X1), X2) → U141(X1, X2)
U141(X1, mark(X2)) → U141(X1, X2)
U141(active(X1), X2) → U141(X1, X2)
U141(X1, active(X2)) → U141(X1, X2)
U142(mark(X)) → U142(X)
U142(active(X)) → U142(X)
U151(mark(X1), X2) → U151(X1, X2)
U151(X1, mark(X2)) → U151(X1, X2)
U151(active(X1), X2) → U151(X1, X2)
U151(X1, active(X2)) → U151(X1, X2)
U152(mark(X)) → U152(X)
U152(active(X)) → U152(X)
U161(mark(X1), X2) → U161(X1, X2)
U161(X1, mark(X2)) → U161(X1, X2)
U161(active(X1), X2) → U161(X1, X2)
U161(X1, active(X2)) → U161(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
U171(mark(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, mark(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, mark(X3)) → U171(X1, X2, X3)
U171(active(X1), X2, X3) → U171(X1, X2, X3)
U171(X1, active(X2), X3) → U171(X1, X2, X3)
U171(X1, X2, active(X3)) → U171(X1, X2, X3)
U172(mark(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, mark(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, mark(X3)) → U172(X1, X2, X3)
U172(active(X1), X2, X3) → U172(X1, X2, X3)
U172(X1, active(X2), X3) → U172(X1, X2, X3)
U172(X1, X2, active(X3)) → U172(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U181(mark(X1), X2) → U181(X1, X2)
U181(X1, mark(X2)) → U181(X1, X2)
U181(active(X1), X2) → U181(X1, X2)
U181(X1, active(X2)) → U181(X1, X2)
U182(mark(X1), X2) → U182(X1, X2)
U182(X1, mark(X2)) → U182(X1, X2)
U182(active(X1), X2) → U182(X1, X2)
U182(X1, active(X2)) → U182(X1, X2)
U191(mark(X1), X2) → U191(X1, X2)
U191(X1, mark(X2)) → U191(X1, X2)
U191(active(X1), X2) → U191(X1, X2)
U191(X1, active(X2)) → U191(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
U201(mark(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, mark(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, mark(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, mark(X4)) → U201(X1, X2, X3, X4)
U201(active(X1), X2, X3, X4) → U201(X1, X2, X3, X4)
U201(X1, active(X2), X3, X4) → U201(X1, X2, X3, X4)
U201(X1, X2, active(X3), X4) → U201(X1, X2, X3, X4)
U201(X1, X2, X3, active(X4)) → U201(X1, X2, X3, X4)
U202(mark(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, mark(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, mark(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, mark(X4)) → U202(X1, X2, X3, X4)
U202(active(X1), X2, X3, X4) → U202(X1, X2, X3, X4)
U202(X1, active(X2), X3, X4) → U202(X1, X2, X3, X4)
U202(X1, X2, active(X3), X4) → U202(X1, X2, X3, X4)
U202(X1, X2, X3, active(X4)) → U202(X1, X2, X3, X4)
isNatural(mark(X)) → isNatural(X)
isNatural(active(X)) → isNatural(X)
U203(mark(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, mark(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, mark(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, mark(X4)) → U203(X1, X2, X3, X4)
U203(active(X1), X2, X3, X4) → U203(X1, X2, X3, X4)
U203(X1, active(X2), X3, X4) → U203(X1, X2, X3, X4)
U203(X1, X2, active(X3), X4) → U203(X1, X2, X3, X4)
U203(X1, X2, X3, active(X4)) → U203(X1, X2, X3, X4)
U204(mark(X1), X2) → U204(X1, X2)
U204(X1, mark(X2)) → U204(X1, X2)
U204(active(X1), X2) → U204(X1, X2)
U204(X1, active(X2)) → U204(X1, X2)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(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)
U212(mark(X1), X2) → U212(X1, X2)
U212(X1, mark(X2)) → U212(X1, X2)
U212(active(X1), X2) → U212(X1, X2)
U212(X1, active(X2)) → U212(X1, X2)
U221(mark(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, mark(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, mark(X3)) → U221(X1, X2, X3)
U221(active(X1), X2, X3) → U221(X1, X2, X3)
U221(X1, active(X2), X3) → U221(X1, X2, X3)
U221(X1, X2, active(X3)) → U221(X1, X2, X3)
U222(mark(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, mark(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, mark(X3)) → U222(X1, X2, X3)
U222(active(X1), X2, X3) → U222(X1, X2, X3)
U222(X1, active(X2), X3) → U222(X1, X2, X3)
U222(X1, X2, active(X3)) → U222(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
U31(mark(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, mark(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, mark(X3)) → U31(X1, X2, X3)
U31(active(X1), X2, X3) → U31(X1, X2, X3)
U31(X1, active(X2), X3) → U31(X1, X2, X3)
U31(X1, X2, active(X3)) → U31(X1, X2, X3)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2) → U41(X1, X2)
U41(X1, mark(X2)) → U41(X1, X2)
U41(active(X1), X2) → U41(X1, X2)
U41(X1, active(X2)) → U41(X1, X2)
U42(mark(X)) → U42(X)
U42(active(X)) → U42(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X)) → U81(X)
U81(active(X)) → U81(X)
U91(mark(X)) → U91(X)
U91(active(X)) → U91(X)
isPLNat(mark(X)) → isPLNat(X)
isPLNat(active(X)) → isPLNat(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)

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